1) Capital ship yard, mines (crystal first), scouts, infrastructure, labs
2) Gain as much ground as you can as quickly as possible. This means usually going for the colony capital ship (more antimatter), wich is in the Vasari case the so called "Space Egg" or Evacuator. It also has some use in planetary assaults later (level 6 as already mentioned), but the ability has quite some cooldown - i really prefer vulkoras later on. Leveling your planet eredicators to level 6 before actual use is a questionable tactic if you ask me and it seems slow. Even if you spread forces thin and you can not really defend all conquered planets, it adds to production and you will soon be able cranking out swarms of LRF (long range frigates, Assilants). I recomend creating a fleet watching its position and adjacent jumps each. Its not a secure wall, but good enough. If you can, secure strategic strongholds - or chockepoints for jumplanes if there are any. Place at least one frigate in every orbit to keep nasty scouts busy and out of your systems.
Then there is always the possibility to salvage the capital ship yard after building your first capital for additional resources early on (you are supposed to rebuild it at a later point, mind this). Whilst this is absolutely great for Advent and useful as TEC i advise against it as Vasari - you do not need the extra early resources since you save a lot on labs on your way to LRF and will gain resources for your next capital more quickly - its simply annoying, giving not much advantage in this case.
There is an exception on small maps where you can sacrifice expansion speed by making a planet destroyer (Vulkoras) your first capital and send it right to the newly founded colonies of your opponent to disrupt his progress early, reinforcing it quickly with LRF and proceed to smash your opponent from the very beginning taking over enemy colonies and establish a beachhead in enemy territory (building repair bays and a frigate factory to take the fight there).
3) As fast as you can. Usually you will be interrupted by a certain stage during your colonization efford where you have to consider either arctic or vulcanic colonies (just look at what type is most common in your reach). The Vasari get usually a good headstart because they need only one weapon lab for their LRF (as oposed to Advent wich need three) and can then quickly proceed to the two needed civillian labs. It takes some resources though and shifts resources away from fleet construction briefly (also research improved ore mining and population enhancements quickly after that, think of it as thrown in bonus, the rest is not really important). Also constantly upgrade your infrastructure as need if population grows on your new colonies, it will soon bring your war engine to full production. Then you should tech up to flaks as a next step and produce like a dozen to deal with any light carrier support your opponent might come up with. Even if he doesn't those are durable ships, capable of soaking quite some firepower. Just be sure to back them up with something WITH firepower, because they are lacking in this category.
4) Trade ports are not as important for Vasari as for other races. Vasari have them on civil tech 4 wich makes them quite... late. I would advise against trade ports as Vasari until much later (much, much, much later) because you need to get up phase missile tech and that is in the military tree. A big no. Go conquer some slaves. Do not trade.
5) If you go LRF or strikecraft (and there are few alternatives) phase missile tech has absolute priority. Especially when facing Advent opponents. Subverters augment that effect. They are late but definetly along the military road. That is the way to go. Just do not be to hasty. Fleet size is more important than tech (except for colonization tech, wich should be achived quickly to add worlds to your empire fast). Usually i build military labs if i need to raise the fleet cap, ordering a capital with the resources piling up during lab construction, building up step by step whilst keeping a good balance.
6) Siege capital. Flaks. More LRF/Strikecraft. More capitals. Subverters.
As far as capitals being concerned Vasari offer lots of variety. You should have your colony capital first (unless rushing as mentioned above), then siege engines (Vulkoras, bombers can pull it off too, but they are slow and vulnerable like colony vessels capitalships get the job done better - it can be advantageous under special conditions sometimes, though). Next is personal preference. If you want strikecraft focus on strikecraft then build carrier capitals mixed with Transporters. Even if you don't two carrier capitals will be beneficial in your spearhead fleet because of the repair ability. If you are cornered Kortuls might provide the firepower in breaking enemy fleets. And even a single Marauder can be a mayor annoyance to the enemy due its invulnerability ability and area-speed-enhancer either for rethreating or as offensive weapon raiding enemy colonies (for an example with an 6Egg to soak up resources and harass infrastructure accompanied by swarm of bombers aming directly at the colony).
7) Difficult to answer because it heavily depends on the opposition you are facing.
By default you need 3 key elements:
-firepower (lots of firepower)
-a way to deal with strikecraft (depending on what you are facing)
-siege power (some, just to get the job done quickly)
Repair would not hurt but it is not really exclusively required.
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