Still new to MP, so I want some help on basics for playing pvp matches. Got a few questions. (Vasari player here btw)
Do I prioritize research in civ or military tech first? I see a lot of people go for trade ports asap, which seems like a good idea, obviously, but where do you start putting into health/armor/weapon techs? Do you build more trade ports than research labs with spare logistics slots?
It really depends on what sort of game you're playing; on how much space there is between you and your opponent. If you're in a reasonably spacious map such as a 1v1 random small, it's usually viable to go for trade before you start fighting your enemy, if you're good at expansion, although unless you're skilled you might have some trouble as Vasari (4 civic labs required for trade research).
On the other hand, if you're playing in the most common online game type, the 5v5 single system, you will almost certainly need to fleet up ASAP to fight the enemy closest to you on your flank. Going eco on the flank in a 5v5 is almost always a recipie for disaster. Even a single civic lab can cripple you, unless you're a very experienced player (or are facing a very incompetent opponent).
So in a 5v5 you will usually want to get one or two military labs as soon as you can and then pump out Assailants or whatever long range frigate your race uses in order to effectively contest territory between you and your enemy; the enemy will probably be building a large fleet as well very early into the game. Keep in mind that every credit spent on research is one credit less spent on a fleet: generally, research as little as possible, except for perhaps the first few hull strength upgrades, and cheap weapon upgrades if you have a lot of frigates that use that weapon.
However, if you're lucky enough to spawn between two allies, you're in the "eco slot" and should focus completely on colonizing planets, building trade ports, and feeding your allies. If you're in eco you rarely want to fleet up early unless one of your flanks is dying.
Overall fleets usually compose of carrier capitals, carriers, repair units, and long range frigates? Should I go for overseers right away and spam those + assailants?
Only TEC have effective fleet-wide repair cruisers (Hoshikos). Advent and Vasari don't have very good options for that sort of thing unless they're defending (and have repair platforms, obviously). Vasari should often focus on Assailants for the first few minutes and then change their fleet composition depending on what sort of ships the enemy fleet has.
Overseers are only that effective in healing *large* targets - but they do that very effectively, and can heal thousands of HP in a second or two if you have a lot of Overseers. They're very useful in combination with an Orkulus, or with a capital ship.
Carrier cruisers aren't used all that often (in 5v5), at least in the early-mid game, because they're relatively easily countered by light frigates, which are easy to build.
When is it best to upgrade fleet capacity, and how often?
(In a 5v5) Once in the very beginning, probably before 5 minutes in, so you can build up a large amount of long range frigates quickly (along with a second carrier capital ship) - and then, once your 250 fleet supply is filled out, you can consider upgrading it further depending on your income and the size of the enemy fleet. It really depends on the situation - some players prefer to focus completely on fleet and military so they can steamroll their opponent ASAP, and others prefer to let you live but spam trade ports to gain an economic advantage.
And something I really need help on, colonizing quickly. Seems everyone else is always a step ahead of me in that department. I try to be quick and use a carrier to clean out most worlds, and then send a new colony frigate in to take it, usually when the last siege frigate is destroyed so I don't have to worry about losing it right away to bombing, but I see others taking worlds much faster.
You do manually target the siege frigate with your cap and strike craft, right? You should be able to kill them quite quickly (and then warp in a colony frigate to colonize). Sending in 4 or 5 LRF can speed this process along a lot as well.
In a 1v1 or 2v2 though, optimal quick colonization usually involves a more complicated strategy: in those sorts of spacious settings, the best players online build a colony capital ship and then build 8-12 scout frigates, to be used for combat. The colony cap warps in with the scouts, and the player manually targets the siege frigates: once they're dead the player can colonize, build a single turret, and move on to the next planet. The whole process often takes less than 2 or 3 minutes. (The turret will almost always wipe out all the remaining militia, except for LRM frigates, which outrange the turret, so you'll need to warp a scout or two back in to kill the LRM).
That's a pretty involved strategy though, so newer players would be better off using a different strategy: a colony cap and a carrier cap can clear planets completely, and relatively quickly. It does require going up a supply level early in the game, unfortuantely.