What he said. I agree in full with Shabbaman's comments. I never automate workers- yes, it's more micromanagement, but the AI just doesn't do it right.
I posted yesterday on a different thread that "In Civ II, I got accustomed to waxing the AI at Deity level with scores over 200%. When I first got Civ III, I figured I could just drop down two levels to Monarch, learn those pesky new units and concepts, and soon be right back up to Deity. Boy was I wrong! The first few times I tried, the AI used me as a Super-Swiffer to mop the floor.

I should have started at a lower level. I didn't move up to emperor until I had 3 consecutive Monarch wins, and even then my first few emperor games were ugly."
Biggest change for me was getting used to nationality and culture. It forces a huge change in how you think.
The advice on this site, especially the War Academy, has helped me tremendously.
The corruption thread by Alexman is excellent for figuring out how to reduce it. It's a far bigger problem than it ever was in Civ II. The number of cities you have is a big factor- 16 is 'optimal' for a standard size map, 24 for big, 32 for huge. Go above that and you're adding extra corruption. One big help is building the Forbidden Palace- it increases the 'optimal' number, and dramatically reduces corruption. Put it in a location that's relatively far from your capital but surrounded by big cities of your own. How do you build a big project like that in a city that's loaded with corruption and waste? You can speed the building process by chopping down nearby forests and by disbanding units in your future FP city to build the FP. Disbanding conscripted riflemen (available with Nationalism and 'draft citizen' in cities of size 7 or bigger) is quite helpful.
Keeping up in tech is hard. I loved the poster who said "Nuclear Weapons? Huh!! My words are backed by spearmen. Fortified spearmen!" For a Civ II techie who liked the builder approach, I have learned to play from behind in tech. It's not easy, but I've had my musketmen taking on a horde of AI infantry and managed to win.
Welcome to Civ III and civfanatics! Sleep is for the weak.