I had played some of Civ I and found it too difficult to play, but I had always liked the original boardgame. When I saw Civ III as my local used game store for $6 I figured I couldn't lose. It even had the original box and instruction set.
My first attempts were pretty pathetic. I tried to be the bully with Germany and failed against the pacifistic Indians who kept on parking in my territory. I later played the Americans and had a whole continent to myself, but seemed to play the whole game fighting off barbarians. In that game I was in the 1800s and still couldn't get a source of saltpeter. I finally gave up on that game seeing that my opponents had totally split their continent and would probably be moving in on the open spaces still available.
I then started reading the articles in the war academy and got some good ideas from there. I chose Persia and actually won a game. Granted I made plenty of mistakes along the way. First off I left a city undefended early as I didn't think there was anyone else out there in range to hit it. That's when a British unit marched in and took it over. Luckily, my citizens rebelled and I had it back in short order. I figured that was the time to march into Britain and teach them a lesson. My military was pretty pathetic, but the Brits weren't anything to call home about. I was working on trial and error to see what worked. Eventually I was just fortifying spaces near his cities to kill production and making the occassion attack with archers until they were dead. It was stalemate until I developed Immortals.
It seemed during the lengthy war that nothing got done, and the two cities I had won had little to offer. The British had even slipped out of the noose by having another city way out in the fringes that I hadn't even explored yet. But the Indians weren't willing to take advantage. When the continent began to fill they asked for a rop which I granted. They built cities between mine which eventually became mine when their citizens saw the bigger and better deal I was offering and came over to my side.
By this time I had completed the Lighthouse and began sending settlers to the smaller continents to the north and west. Their production never really took off, though I would say that overall they had a slightly positive effect. A big mistake I made there was in not putting a forbidden palace in one of those cities.
So with the Brits never really able to take off, and any new Indian cities being ceded to me, I controlled a good chunk of the main continent and was making inroads to the New World with heavy production in settlers from the old world. Ferrying them across the sea was a slow process but it was eventually working, especially when I found iron there.
I was in the 1900's at this time and still had not dominated despite the fact I had the New World to myself and my cities covered a good deal of territory. I noticed the Indians were still using archers and spearman along with their war elephants, while I had calvary and infantry. I still did not have modern tanks or planes, but knew that my units were superior to his. The oil well on his land that he could not use sealed the deal.
I practiced by ferrying some troops to take out the one remaining British city. Pathetic still running into warrior units in the 20th century. I attacked the Indians meeting very little resistance in the unconnected cities he had to the north and the newly developing cities to the south of me. About the time I was taking on his city near the oil reserves, my cities were in revolt, along with the cities I had just taken over. I captured the city near the oil and sued for peace. I started some tank units, but the oil withered within one turn. Since I had started the units I was able to produce them, but could not start anymore or upgrade my infantry to mechanized.
The oil was again in his territory, so I struck with my new tanks. They were of limited effect. The majority of the battle was taken to the enemy by my calvary units, which meant I was open to counter attack and moved slow allowing my infantry to keep up. By the time I reached the city near the oil fields, all my cities were in revolt. I had to quickly take the town and again declare peace.
The Indians were just a shell. To top it off, I couldn't use the oil because my newly aquired cities no longer had the coverage. I had to build a colony there just to take the oil. What had the Indians been doing for 6000 years. The cities I had taken over did not even have temples. I am guessing they probably sold off the improvements when it became clear I was going to take them over.
After quick rushing some improvements on my newly acquired land I finally had enough territory to win the game. I played on for a bit, but it became too frustrating when the Indians sent a settler to the New World and built in where I didn't have coverage yet. I couldn't simply boot them out at this point.
I'm thinking my next game will be either Japan or Babylon. I'm working out strategies for the various strengths and want to try a religious society. I love industrious as you can really set up the land with even a small group of workers available.

Since I have just played another game and won with the failed American strategy I laid out, I thought I would go back and look at this game and look over this game and see why some things seemed harder and some easier. Feel free to offer some constructive criticism, I will be doing so myself.
I have the vanilla game and am playing the Persians on chieftan level, standard size map with continents, normal climate, 4 billion years old, temperate with barbarians held to villages. My opponents are the Indians and English as I wanted to stay away from warlike opponents.

Looking at some things right off the bat I can see some issues. I hate running workers and will often set them to auto, but this one seems to be making some idiotic decisions. Why didn't they build a road to the gold with a mine there? It would have also made more sense to put in irrigation in the flatlands instead of mines. At this point it looks like the computer is trying to balance off shield and food production, a mistake IMO.

Its still early and I have yet to meet the neighbors, so I have left my city undefended to aid in exploration. I have set science to 50% and have developed bronze working, masonry, the wheel and warrior code, some with the assistance of friendly tribes in the area. Pottery will be available in six turns.

Couldn't get the save in because the file is too large. Don't know why that is, one would think the save files are all the same size or smaller in a newly started game. I will work to make future images clearer as well.

I don't really have any game-specific advice for you, but I do have some general advice; I found one of the most helpful & resourceful entries in the war room to be Deity Settlers (http://www.civfanatics.com/civ3/strategy/deity_settlers1.php) by Bamspeedy. He is playing on Deity level so things are a little different, but it helps one plan for the worst. Also, a lot of the information is seemingly unimportant because of the small effect/difference they make, e.g. The micromanaging of the 2nd citizen to get an additional 4 shields before switching it to a entertainer, but it is very important on higher difficulty levels.

I'd say the most important aspect about Civ is what you know; about the mechanics of the game, the map, the AI tendencies, effective use of artillery, what tiles to improve first, which Civ is your biggest threat or best target.

So I guess I'd recommend you just keep playing, and reading the war room entries, it all comes with time :)

Good luck!

Couldn't get the save in because the file is too large. Don't know why that is, one would think the save files are all the same size or smaller in a newly started game.
Are you using an autosave or a manual one? Manual saves are smaller, I believe.

I don't have an autosave on. I use the little triangle at the top of the main menu to save. All my save files are about double what is allowed. ??? Should I just hit control-s instead? Are there certain preferances that might be taking more space?

I can't think of anything in the preferences that would do that. . . I'm not very good with tech support. I just know that's a pretty common reason for saves to be too large. AFAIK, it shouldn't make any difference whether you're using ctrl-s or the little triangle.

Time to jump ahead to 1375 BC. I had met the Indians without incident. The Brits on the other hand had introduced themselves by sneaking into my second city which I had left undefended. Luckily my citizens rebelled and it was mine again in a few short turns. I decided to strike back at her cities and take them out of the game. My plans were not moving along as I had no plan to win. I was using defensive pieces on the offensive. Luckily the Brits were not in a position to strike back. My strategy to whittle them down was to fortify important production tiles and deny them the ability to strike back. When they did strike back they were running into defensive units in fortified positions.

As you can see my treasury is not doing so hot. I've developed horseback riding and iron working, two techs that should have put this war at an end. I was too inexperienced to take advantage. I have the alphabet and am working on writing. If I had seen the horses near Des Moines...

Its 150 AD now and still at war, but here come the immortals. Notice that the Indians have moved into an area in which I will need to expand in the future. The fools...

As you can also see I had not learned to stack and wear down an enemy. I did not know it at the time but the Brits had saved themselves from extinction by building a city at the other end of the continent. I would be at war with them even though we wouldn't run into each other for centuries. The treasury is starting to pick up with new cities and I now have access to iron. I'm at least doing some things right.

Its 350 AD now and I must have let the power go to my head. The immortals are on the move again and my capital is building the pyramids. Looks like I still haven't gone after those horses near Des Moines. The territories I have conquered are not progressing very quickly. My ideas of city development were not, erm, developed. In hindsight I should have built a forbidden palace in London. The treasury is progressing, but perhaps not as quickly as it could have.

By 570 AD and not much has changed, so little that I won't even bother posting a picture. I'm still a despot and my treasury is a little richer than before, about double what it was in 150 AD. Just before I wrote this I was playing around with the save from 350 AD and was able to get into Delhi within a few turns. I didn't even notice that I was building granaries in towns when I already had the pyramids. I didn't even think that was possible. I'm still stuck in the ancient age with no currency tech, polytheism or any of the advanced governments. I am behind India on the histograph but well ahead of the Brits whom I have pretty much eliminated from civilization.

760 AD and still at war with the Indians. It doesn't even look like taking his capital is going to be worth it. But trudge on I do. At least I have horses now, but not much to do with them. I wasn't aware I could wear down the enemy and then move in. I'm still a despot and still in the ancient age. Even if I take Delhi India has a source of iron. I have not built any new settlers but did overtake the Indians for the leading score, only because now I hold the most territory.

By 870 the future is looking brighter. Delhi is taken, the treasury is building and the Republic is less than a century and a half away. Still not all was budding. I was still trying to advance with inexperienced units while elite units are left guarding the rear. I still had not noticed that two of my cities are unecessarily producing granaries. I have been at war for more than half a millenia, more if you count my battle against the Brits. Good thing I'm a despot, and a bloodthirsty inept one at that.

By 1060, or historically about the time of the Battle of Hastings, the Republic has taken hold and my citizens are tired of the constant warring. I have fallen back from Madras and somehow one of his units slipped past me and is threatening one of my cities. Good thing its only a spearman. My cities are in civil disorder and its time to go with the ways of peace. Marion was in competition to build the Great Library with Madras, but Gandhi changed his mind. Most likely because the city had been taken to a low point in population. I guess I didn't realize how close I was to making it obsolete. Corruption in my cities is equal to almost what I am putting towards science. My cultural score is just 2127 but I am starting to pull away from India on the histograph with a score of 117-83. England has plateued at 36. I am still in the early middle ages with work on Feudalism. Surprisingly my my cultural building is the Temple of Marion.

My next save was not until 1335. It looks like despite warnings on the War Academy I am heavy into wonder building with work on Leonardo's Workshop, the Oracle, the Forbidden Palace, and the Great Lighthouse. I was building the Forbidden Palace in Burlington though it probably would have served better in London. India has begun building in the space between the cities I've built. That is probably because much of the free space left is tundra (yeah, you know the oil will be there later). I notice that I hadn't even taken advantage of the gold deposit right outside of Marion, no road or mine in that hex. I am starting to pull ahead in civ score, but most of that is territorial gain. The Brits have only 3 cities of which Nottingham is the largest with a population of 2. Amazingly enough they are trying to build the Great Lighthouse even though it will take 136 turns. They haven't even built a road yet. I have also begun to pull ahead technologically from both of my challengers.
The city of Perseopolis was founded in 1200, my first new city since 835 BC! It was also my first coastal town and would serve as the focal point to reach the new world. The cities I have taken from India are taking off except Kohlapur which I did not notice has its only citizen as entertainment.

The new world fielded its first town in 1465. This shot is from 1530 AD. There are not a lot of luxeries or resources. There are some wineries in this shot and some gems would pop up to the west, but they were in a mountainous area that would not allow for growth. I would have trouble getting those cities to produce. That was probably in part because of corruption and also poor town placement.
Well at least the other nations haven't seen it yet. My cultural score was pretty low, just 8346 after more than 5500 years of civilization. When you go from a war economy to just trying to get along, culture takes a back seat. In addition my automated workers are not really helping. Makes the turns go faster, definitely, but you probably need twice as many to just do an adequate job.
On the good side, Gandhi has been building new cities in the middle of my territory and they've started rebelling over to my side. My sliders are still set to 50% on science and 0 in lux. Technologically I'm about halfway through the middle ages.