Instructions
Colonize nearby worlds. The closer a world is to your capitol world, the less culture decay that world experiences. You reduce culture decay early in the game by only colonizing planets near your starting world.
Build improvements. Each race has several improvement structures they can construct -- such as the "Broadcast Center -- that will increase the culture rate on outlying worlds. Build these structures to keep your culture from decaying.
Build capital ships. Each race has several different capital ships to construct. These capital ships, when in orbit around a planet, will increase that planet's culture simply by it's presence.
Change your "Capitol Planet." If your empire starts to expand too far beyond its original borders, you can shift the capitol to a more central location. Doing this will decrease culture decay for all the planets surrounding your new capitol.