Thanks for all the recommendations. So I built an exploration fleet with lots of fuel and cargo capacity, and made really good money doing that. Now that I've upgraded to some good combat vessels (3 cruisers (2 Eagles and a Dominater(?), a mix of 3 destroyers, 3 Moras, and 6-7 support frigates that I usually set to escort during combat. They are mostly Wolfs, but 2 Lashers, and the custom frigate). My support ships are a couple of tankers and cargo ships. I just bought the really big cargo ship (Atlas?).
Now my problem seems to be that I literally can't find enough fuel (and sometimes supplies) to do much. I can buy everything, but maybe I need to visit different systems and buy all their stuff? The bases are literally out of fuel and supplies. It's not that I can't afford it.
Maybe I grew my fleet to big too fast?
I had to Mothball half my ships to get back from my last mission without running out of supplies.
Thanks for the tip on solar shielding, too. I might try that with my exploration missions.
So, the fights are getting better. The pirate fleets are really scaling up, though! I've managed to siege and destroy a couple of their bases with this fleet and not take too many losses. The problem is that some systems will have so MANY pirate fleets. I could defeat them singly, but they all try to chase you and gang up. 
I'm thinking about selling/storing some of my big military ships and going back to exploring the rest of the galaxy. Maybe try to start a colony?
Fuel - Try Sindria, Nachiketa and Madeira.
Supplies - Try Chicomoztoc, Asher, Kazeron, Culann and Nova Maxios
Maintaining a big fleet is useful if you're planning to do a lot of combat, but if you're only exploring, then you just need a few ships that can punch above their weight class. Right now, I'm running around the Sector with 2 Auroras and 2 Scarabs, and that's literally enough to fight off basically any threat I'll come across. The rest are just logistics ships.
You will encounter some systems with a lot of pirate fleets, or a lot of [REDACTED]. Either you find ways to pick off one at a time (Go Dark is a nice option for this) or you avoid the fights altogether until you can build a fleet that can take down 8+ fleets in one engagement. For that, you will need capital ships, a very optimized fleet of cruisers/carriers, or to change your name to CapnHector.
Starting a colony is one of the joys of Starsector, but it can also bring with it a lot of...responsibility. I would counsel you to at least spend some time getting ready for starting a colony, because things can get very hectic once you do. There will be a LOT of things you have to manage, and it opens up parts of the game that you haven't even been exposed to yet.
My rough checklist for whether I'm ready to start a colony:
- $5M+ in capital
- A system with planet(s) I'm interested in colonizing
- Special items that synergize with those colonies
- A Starliner
- 1000 crustaceans
