When I answer these questions, I'm going to answer them off the top of my head. I'm not going to pretend to know things that I don't, and so I'm not going to give any textbook answers.
This isn't so much about being exactly technically correct, as trying to impart the amount of knowledge that is required to master the game. I don't want to leave anyone thinking that to be really good you have to know the cost of every technology off the top of your head, or anything silly like that. I know that there are a few players who are 'experts' in technology cost, but most good players just know the basics. Same with most things.
My answers are as follows:
1) The Greeks are scientific and commercial. Being scientific means that libraries, university, and research labs are half price. (research labs come so late in the game that they are of little consequence though). We also get a free technology at the start of every age. This technology is usually the top-left technology on the technology tree: monotheism in the middle ages, and nationalism in the industrial age, and I forget in the modern age.
Being commercial means that we have reduced corruption. The corruption formula is actually reasonably complicated, but we will find corruption reduced by a substantial amount. This corruption-reduction effect used to be almost unnoticable, but they doubled it a couple of patches ago, and now it is noticeable.
Additionally, the city center tile now produces more commerce than it would normally for cities above size 6. For cities size 7-12, I think it produces one extra commerce, and three extra for cities size 13+, but I'm not sure of the exact numbers.
Needless to say, being commercial means we will find ourselves with substantially more commerce than we would otherwise have.
2) The Greeks start with bronze working and the alphabet. When considering starting technologies, think about it logically: religious civilizations are going to start with ceremonial burial. Militaristic ones with warrior code. Expansionists want to expand, and granaries help with that, so hey have pottery. The Wheel and Masonry fits industrious civilizations. The alphabet matching up with commercial and bronze working with scientific isn't quite so intuitive in my mind, so you just have to learn it.
It is not uncommon for the alphabet to be traded for two other starting technologies.
3) The hoplite is the Greek unique unit. It's 1/3/1, replacing the 1/2/1 spearman. It requires bronze working to be built, and costs 20 shields, and doesn't require resources, the same as the spearman.
4) The first observation is that we are not expansionistic, so we don't get scouts. This means we will have to explore with slower units.
We do get hoplites from the very start, since we start with bronze working. This is a good advantage, but it also leaves us with difficult choices: Do we build warriors or hoplites to start with, and explore with? Certainly if we wanted the military for fighting, hoplites would be a better choice; but early military isn't just for fighting, it's largely for exploring and military police, and warriors do this just as well as hoplites at half the price.
An aggressive/expansionary approach would be to build 3 warriors first up, with two exploring, and one guarding the capital and acting as military police. Once settlers start getting built, hoplites could be produced to guard our cities at that stage.
A slightly more conservative approach would be to build 2 warriors and a hoplite, with the hoplite guarding the capital, and the warriors exploring.
These approaches will also depend on in-game factors though: How near other civilizations are, the presence of barbarians, the food/shield balance of our capital, and so forth.
Having hoplites means we should be relatively safe from attack in the early stages of the game.
Besides determining our starting technologies, our civilization traits don't do much for us early on. Being commercial won't kick in until later on, and we won't be able to build cheap libraries for a while.
5) We definitely want to discover pottery first. No matter what. It's a cheap technology, and it gives us access to the granary, which is far more powerful than any other improvement or unit this early in the game. The only possible reason you wouldn't go for pottery immediately if you didn't start with it is if you want to go for an early pyramids gambit, in which case you'd go for masonry. But that's prohibited in this game anyway.
After that, things are much much more up in the air as to what we should do. If we immediately went for iron working, we could probably get it before other civilizations, and thus trade it with them. It'd also help us to secure a source of iron early on.
We also want to try to get literature and republic fairly quickly, as changing from a despotism to a republic asap is almost always a great move, and getting literature will let us start building those cheap libraries and really get ahead in science. So, we could start going that route.
Really though, pottery is set in stone. Beyond that it's much more of a 'see as you go' mentality.
Some comments on other's answers
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Matt_g: We'll go for pottery and a granary no matter what. This is almost always the best option.
Good points about the hoplites influencing our opening, as well as having the alphabet putting us in a good bargaining position. Trading it will require skill: we don't want to give it away, but we don't want to end up with them getting it from another civilization or researching it and us getting nothing. Ending up with two technologies for it would be ideal, getting a technology and some gold (20+) would be good too.
Note that commercial also gives reduced corruption, which is generally seen as the bigger advantage. The extra commerce in the city center starts kicking in at size 7.
Note that scientific also gives the free technology at the start of each age.
Renata: Hoplites cost 20, same as spearmen. A Unique Unit that costs more than the unit its replacing is always dubious, since unless it has substantially improved stats, it could well end up overpriced. Before Play the World, no unique unit cost more than the unit it replaced. In PTW, they ran out of unique units to make, so they started making 'super units', and then pricing them highly to make sure they weren't overpowered.
Carthage's Numidian Mercenary is 2/3/1 but costs 30 shields. In my view, the extra attack point is useless, and I'd prefer the 20 shield hoplite any day.
cgannon64: Good guesses on the technologies
Also you're completely right about the pottery/granary thing.
If there is one thing I want people to learn about this phase of the game, it's the power of granaries and maximized food.
-Sirp.
on your commentaries, Sirp. In the monarch game you inspired me to start I am winning the histograph (considerably larger nation than the others- your settler first mentality really helped me secure land), even on technology (at education/gunpowder), and with a big enough army that when I tell them to leave or declare war, they leave. THANKS!!!
, but I'll put in my two cents. I will always irrigate a bonus food in the early going unless either of the following applies:
