Now that you have a stable economy, make transport ships, then launch them toward different planets and moons. Make sure you check the planet first for oxygen if you're playing as a race that needs oxygen. If you are, and the planet has no oxygen, build a Planter ship and crash it into the surface so that oxygen will be produced. I always like having forward bases that are close to the sun, and devoted entirely to one resource type, like this planet.
It is very useful to be playing as a race that doesn't need oxygen in this part of the game. You can colonize planets that races that need oxygen can't, and if they try to land, they quickly die. Place on every planet at least one spaceyard, so that you can colonize faster, and build warships. During this part of the game, you should build up your army, composed of Transports and Hunters. If the enemy has a moon, it's a really good idea to set up gattling cannons on the moon, to shoot at ships that cross its path. Idea attributed to Orteil.