For the original GalCiv, invading a planet requires that you first research Impulse Drive. This will allow you to build transports.
Build a transport on one of your planets near the planet you want to invade. Upon completion, launch it. You will be able to load it with soldiers, in a manner similar to loading a colony ship with colonists. Then, move the transport next to the system where the planet you want to invade is located.
Use your non-transport ships to destroy all enemy ships in system orbit, then select the transport and right click on the system. You will then be taken to the invasion screen.
I have no idea how to use cheat codes in GC1. I prefer to play honestly.
You cannot go beyond the starting galaxy. If you are referring to the Multiverse, it is, essentially, a high score list. You can submit your game after it's over.