I can't anwser all you're questions, i'm fairly novice myself, but!
[Ship Mod,
Turns standard shields into omni (full coverage?),
But what does it mean by reducing the arc?]
Omnidirectional shield make your shield be able to point in any direction, as opposed to being locked in front of you. The mod trades off how wide forward-facing shields usually are in exchange for letting you point it wherever you damn well please.
[Peak Active Performance Time,
Is this static based on the ship chassis?
Is there a way to upgrade or modify this?]
It's dependant on the ship itself, though there are average numbers for the diffrent classes of ships As for improing it, the only way is with a ship mod, and/or player skills.
[I started off trying to trade,
but couldn't because the contracts were unreasonable.
Almost all were beyond my CR to buy the supplies, and beyond the carrying capacity of my fleet.
The contract requirements just seemed way out of scale. Am I wrong?]
When you first start off, contracts can be a little.. overwhelming, indeed, but some of the bigger ships can have 500+ cargo capacity, while still being otherwise decent, and if you have a few of them...
[Thus, I resorted to killing some weak pirate fleets.
Was this the right/good way for initial progress?]
I don't like telling people that there's a 'right' way to play, but, being said, unless you want to befriend the pirates, they are more or less there to start you off with fighting, even to the point of having crippled ships (D-class)
[Eventually I got myself a cruiser, and then upgraded my support carrier to a Heron.
While the fighters "seemed" nice at first with their point defense capabilities, the carrier + fighters just seem squishy and ineffective given the cost of their deployment points.
I had 2 Xyphos wings.
Am I wrong that they don't seem worth their cost?]
Big o' can of worms that depend on who you ask. I kinda like xyphos wings, but afaik, the general opinion is with the recent changes, Fighters could use a little more love, balance-wise
[Delivery contracts that fit my storage size still seem like a miss.
The contracts are for way too few items.
And then most of the time, there's no easy to way to accept the contract, locate the item and still make the 10 day delivery deadline.
Rarely ever is that item located at the current location (in sufficient quantities).]
See? now you have the opposite problem! hat's the glory of RNG. Atlas is a bit overkill, but it does get the job done damn well for taking *** from point a to b. As for filling contracts, as I see it, if the stuff they needed was already at the place they where posting the notice, they'd just buy it there for cheaper. I find them most handy for getting rid of loot that I aready have in my inventory, or I'm close to having enough of.
[So with me only finding it worthwhile to hang out near my 3 friendly systems (where I can sell my salvage),
and trade being hard to find.
Is this it? Am I at the endgame?]
IMO, the 'endgame' is when you're done with the game. Aside from that, you do what i do mostly, even though i feel guilty because there's a whole sector to spend my time fair-er-ing in
See next anwser for why you should spread your 'wings' a little
[Also, I can only find the Onslaught being sold.
How do I acquire the other capital ships like the Paragon and Conquest?]
Conquest you MIGHT find in the starting sector, but it's unlikely. Since Tri-tach moved shop, the paragon is likewise almost impossible to find in the starting sector. The other regions of space are home to a variety of diffrent groups who sell diffrent things. Indies and dictat are your best bet for Conquest, and tri-tach is your best bet for a Paragon.
[And finally.
What next? Is there more to vanilla I should be doing?]
Aside from building up, leveling up, and exploreing more of the sector, makeing new friends and foes as you go, (I always try to be friends with everyone, but having a foe really makes the game have more flavour, as it makes that area of space a lot more interesting). If you want to get into more fights, a Commision is a good place to look, though I'm sure you already have one if you can get onslaughts.
[And then where should I start exploring with mods?]
This is really up to personal prefrence, espcially depending on what you want from the game!
The mod I use the most is StarSector+, with all of the subfactions. (Actuially, it's more like Lazylib, and ConsolCommands, but shhhh)
The other that I used, though I think it's still broken with the latest update, was Project ironclad, a TC mod that makes the game fairly diffrent!
And if you want huge replayability, I'd be remiss not to mention Nexerlin, which offers a great deal of risk-like gameplay to Starsector, as well as randomization, and capturing.