I'd like to make game, where one part is managing planet (building cities, factories, roads, etc.) but I'd like to avoid 3D - only what I want is nice 2d. But if I represent planet's surface as a grid, it will be strange to move near poles (if I go up, I won't appear in the bottom of the map, but in the top - only in different place) and it will be hard to have nice terrain (poles will be stretched). Do you have any ideas how to do that?
Games should not be realistic, you can always cut some corners if it makes the game better. Take a look at Ascendancy for example:
You can loop the 2D terrain from left to right and from bottom to top. Like a Civilization map does.
Another option would be to make "unwalkable" polar caps, like we have here on Earth ;-) They will hide the areas of stretched grid.
Section 2 of this paper has some interesting notes on hierarchical triangle models, but you might find more immediately useful material looking at geodesic grids. You need to supply the connectivity information, but that will get you to a good approximation of a sphere.
This paper also has some interesting ideas.
As Krom pointed out, you certainly can setup your own system for connections, allowing for colonization of weird extra-dimensional spaces at the most weird, or simpler structures such as a Halo-ish ring world.
The "proper" way to represent a planet's surface without any special treatment of the poles or any other extreme points is a triangulated irregular network. From there, you can use any of the usual geographical map projections to represent whatever part of the surface you want as a 2D map.