Conquest is a game mode similar to Galactic War, where you advance through various nodes with the teams you have built. It has different difficulty levels, initially you will only be able to do easy conquest.
Here is an in-depth guide that answers most of your questions, like rewards, and gear level requirements for each sector.
Each set of 3 conquests (3 months, one per month) has feats that require different characters and/or factions. It can be difficult to farm characters specifically for conquest, as by the time you farm them conquest will change and you will need different characters. You can see the feats for the upcoming set of conquests
here, check the section for Easy conquest.
I would suggest building a farming path of building core teams of different factions, while you work towards your desired objectives, like a Galactic Legend or a high level fleet. That will help you in conquest as well as in all game modes.
