95% of the time, your first capital ship should be either a colonizer or a carrier (for Vasari, that's the jarrasul or the skirantra)...this is true for all factions, not just Vasari...generally, colonizers are better for more open maps (where there's more space per player) while carriers are better for more cramped maps (where there's less space per player)...

You should always focus on colonizing planets quickly, with the general rule of expanding towards your enemy...Sins heavily favors horizontal growth over vertical growth...

Best techs for Vasari really are the best techs for any race:

• Colonization techs (the ones that allow you to colonize ice or volcanic planets)
• Prototype techs (ones that unlock ships or structures) - in general, getting long range frigates (LRMs, kanraks, or illuminators) and flak ships (gardas, sentinels, or defense) as soon as possible gives you the most options for your early game fleet...repair structures should always be gotten early...trade ports are usually the best economic structure to get, and you should start spamming them once you get 4-5 planets
• Preferred weapon techs - this usually is the weapon techs that improve the firepower of long range frigates, light frigates, and flak ships...specifically for Vasari, that would be phase missile upgrades (and possibly pulse guns, but just phase missiles is fine)

Most economic techs in Sins are useless unless you are playing ridiculously long games...the pay off times just are too long compared to the length of a typical game...techs that boost trade income (for the Vasari, Pricless Goods) are about the only one's worth getting, with the occasional benefit from terran population bonuses...