How to upgrade the mothership?
All Mothership upgrades are locked behind story missions, so it's entirely progress based. They are:
Rebuild a fleet (part of the tutorial) - the mother ship gains max capacity (500 resource, 50 goon) and expands the Hanger to have two slots: small and tiny.
Finish Repairs (also part of the tutorial) - 2500 Rez, 150 goons, can have 4 specialists with 1 active. Hanger slots: Large, Small, Small.
Big Fish - 10k rez, 150 goons, 8 specialists with 2 active. Max Tech level now 8 (from 6). Hangar: Huge, Medium, Small.
Pandora's Box - 50k Rez, 1.5k Goons, 12/3 specialists. Max Tech level 10. Hangar: Huge, Large, Medium, Small.
Pandora's Box spans the entirely of chapter 3 in story terms. It has many steps. If one skips the tutorial, the game starts with the Clockwork upgraded as it would be at the end of Finish Repairs. Also note that in the demo there are a few tweaks to the upgrades (e.g. Finish Repairs only gives a medium hangar slot).
The mothership is upgraded during the story. You have to advance further to get more hangers etc.
As for getting further one easy way I've found was to jump into high level sectors(bribe your way there) where you are neutral to both sides and then find conflicts and pick up blueprints when they destroy each other. You can easy pick up some higher tier ships that way.