>>2116731
To begin, go to one of your colonies. Approach the planet or its station and click "Special functions" at the bottom, then "Request a fleet". Set the mission type to invasion, the origin market to one of your planets (I'd suggest one with a high military rating, like a world with a high command), then the target market. Upping the task force strength will cause your faction to send a larger fleet with more marines, but if you have a huge fleet and a lot of marines already you can afford to send a cheap force so long as you escort them closely.

It'll take some time for the fleet to form, so hang out at the planet the fleet is launching from. When it launches, have your transponder on and follow behind it closely - don't fly in front of it or too close, or it'll try to maneuver away from you (some sort of old code to avoid fleet collisions I guess), and if your transponder is off it can possibly mistake you for a hostile fleet and try to flee if you're close enough to detect but too far away to get visual confirmation on.

When the fleet reaches its target, it'll take about a day to deploy marines and begin the surface battle. If *you* invade the planet before this then it'll cancel the invasion and give the planet to your commissioned faction if you win, so don't do that. Instead wait for your faction to land troops and begin the battle, then open the surface battle screen, join your attacking forces, and land your own troops. When the battle concludes, your faction should conquer the planet in your name.