Civ 5 always gives you a decent starting position, but I like to tweak it if possible by repositioning for one or more of the following:
1. River - allows you to build the watermill and hydro plant, plus you generate extra gold in trade routes.
2. Isthmus - on a continent or decent sized island, cutting down the time required to circumnavigate the landmass makes your ships much more efficient.
3. Hills - sometimes I move onto or off hills depending upon what my long term plan is. The extra hammer allows for a faster start and the defensive bonus is great, but the windmill is a fairly important mid-late game building and and being on a hill locks you out of it.
4. Mountains - if you can reposition so that you're within two tiles of a mountain, it opens up two wonders for you to construct. If you reposition next to one, you can build the observatory, which is insane when combined with the science national wonders.
5. Marble - If the game spawns me with marble slightly out of reach, I usually move closer to it so that I can get as much out of the bonus to ancient and classical wonders.
6. Civ bonuses - Celts, Huns, Japanese, Russians and Spanish all get bonuses from specific tiles, so if you can position these within 1 or 2 tiles of the city you can exploit them really early on. Then there are unique buildings unlocked later which benefit from optimal positioning, such as the Longhouse (near as many forests as possible) or the Floating Gardens (extra food from lake).
7. Coast - The game is usually good with coastal placements but I find that sometimes you can tweak it by moving up or down the coast to increase the amount of land squares you can work.
I never spend more than three turns moving though, even two is pushing it.