Should I be looking to glom every wreck that doesn't require a story point into my fleet until I hit the maximum fleet size (I assume there must be one)? Save up a couple hundred thousand credits to buy some of the capital ships I'm seeing in the black markets now that I'm out of the tutorial?
There is a fleet size cap of 30 ships - you can go over, but get pretty hefty penalties for doing so. While capital ships are tactically powerful, they're also likely to slow down your fleet a lot (slow burn speed) and can be pretty expensive (and resource hungry) to maintain and thus strategically not a sure thing. I'd advise sticking to cruisers and below for a while, ideally trying to keep your burn speed relatively high so you can hunt down other fleets effectively (and be able to avoid being hunted yourself). I try to keep it at least 8 or 9, though 7 isn't terrible.
I realize that the game's a sandbox, so the answer is probably "whatever you want" - but, without getting too min-max'ing about it, what are some good (or very bad) ways to start out in the universe?
The latest version does have a main storyline/plot:
Visit Galatia Academy to get that started.
The storyline does include a set of RNG/repeatable quests and is not time sensitive, so feel free to dip your feet in a little for some initial employment opportunities.
Outside that, you may want to pick up a commission for steady income, though your employer's conflicts will affect your own hostilities as well and so can be liability. Colonies require a significant investment but will provide a very solid base of operations, so doing at least some exploration for good candidates is a good idea. They also take a while to get running, so starting one earlier rather than later isn't a bad idea to get that population growth going. Doing that exploration as part of a mission (e.g. scanning a derelict or the like) will help keep things more profitable.
Colonies also offer free storage, which is nice - there are a few scattered locations in the core that can also serve this purpose, though.
One of the more popular mods, Nexerelin, changes the relatively static map into something closer to a 4x conflict - I wouldn't suggest it for your first game, but it's a great addition for a subsequent play.
Edit: also - what should I be doing with all these weapon systems I'm picking up? Is there an easy way to tell what would be upgrades for my fleet, or do I need to individually check each one against each ship mount on the entire fleet?
Latter, pretty much. There is an autofit feature you can use, but it's not 100% reliable. Over time you'll get a good sense of which weapons are useful and which aren't, and which ships are doing decently v. which aren't, but it takes time. E.g. railguns are really good, and ideally should be replacing autocannons if the ship's OP can support it.
One minor tip on that point: while the simulator is good for testing duels, one-on-one combats are only a small slice of things. (The simulator will let you use your other ships as allies, but that's still a relatively limited slice.)