I use Babylon because I prefer religious and scientific for the quick government switch and the free technologies
On any level go for the great library if there are many civilizations in my first city other than the capitol i build the palace until I can get the great library
For my play style (relatively peaceful builder, but perfectly willing to go to war if I feel the need) on the maps I play (huge archipelago) at up to Regent difficulty (and I'm pretty sure at higher) Persians are AWESOME. Industrious is, IMO, the best attribute by far, and Scientific is the only other one I've found the least bit worthwhile. (Expansionist is totally useless except for the first 10-15 turns, Religious isn't much good because I very rarely change governments, I don't fight enough for Militaristic to be worthwhile, and since I have yet to have substantial problems with corruption, Commercial doesn't seem too useful). Also, their unique unit is really extraordinary if you get involved in an ancient war - which is where I find I need the extra edge.
Persians Gotta love 'em - in a platonic sense, of course.
I think Industrious is the best because of the improved worker performance. The extra shields also help a lot, especially in the early game, but the increased worker performance means faster infrastructure early in the game then cheaper maintenance costs (fewer workers needed) in the mid and late game.
Expansionist is very good on the large or huge maps, you will discover most of the the ancient techs in the goody huts and get to the middle ages very quickly. That gives you an important tech lead early in the game. That plus the Industrious workers should keep you ahead. On smaller maps, expansionist is not as valuable.
Religion is good for building those cheap temples early in the game. After that, it doesn't help me a lot because I never switch governments after I get to Democracy. (It also lowers the cost of cathedrals, which is good).
Science means cheaper libraries, which is almost as good as cheaper temples. Cheaper universities and research centers may push science ahead of religion in the building race. And three free techs in the game is nice but it would be a lot better if you could choose them.
Military means cheaper barracks, but they are relatively cheap already and if you build Sun Tzu's war college then you already have barracks. Later military improvements don't really need a production boost. More leaders is the other advantage of military and this will be important for the habitual warmongers. But if you prefer short wars and long periods of peace, then military is not so good.
Commercial means more money and less corruption. The benefits aren't as obvious in the early game, but on a huge map with many cities it could be a good trait to have.
With that said, the best choice seems to be the Americans. However, I've played the others and I think none of the civs have overpowering traits. You can win with any of them. The AI Egyptians seem to do very well in every game I've played (I especially like the Egyptians because their War Chariot is a pushover for my Knights, but they do very well against the other civs and will be a good ally until you're ready to squash them). The performance of the other civs has been mixed. I think good terrain and position is more important than the traits.
The unique units tend to give the civs a boost and are timed to coincide with the period that each civ was historically strong in. But the UU's are not overpowering and can be countered. Besides, the AI seldom capitalizes on its UU.