What should a typical opening be? Are the Vasari good at economic openings like the TEC or is it better to go for more aggressive openings? What are good starting capital ships?
Both the TEC and Vasari have solid rush openings, but the TEC have a better economic opening. While the Jarrasul colony capitalship is better in combat compared to the Akkan its colonizing bonus is far weaker. Even you decide to get one though it will come in handy later, just don't expect it to kick start your economy in quite the same way.
The other main capital choice is to go for a Skirantra for the healing/early bombers (it can take out terran and desert militias by itself thanks to this) or if you are close to your enemy, a Vulkoras siege rush can work well against the AI or players who don't see it coming.
What should a typical opening be? Are the Vasari good at economic openings like the TEC or is it better to go for more aggressive openings? What are good starting capital ships?
The main weird thing about the Vasari is that they get their long range frigate with one lab and their corvette at two, the complete reverse of the TEC. I still usually go 1 mi - 2 Civ ASAP as Vasari but just know this means you'll be building LRFs, not Corvettes, early on.
I’ve heard a lot about Vasari starbases being vital to their strategy, how soon should I attempt to get these? Should I use them defensively to block chokepoints to free up my fleet to attack or should I use them aggressively within enemy gravity wells?
If you want to play like a pro, as soon as possible. 
The art of rushing a Starbase to deploy on the enemy homeworld is a unique Vasari tactic that everyone that wants the master the race should be able to pull off, even if they don't do it every time. Even if your enemy fights it off eventually, it usually stalls them far more than you. If you go for this Strategy, a Skirantra as your first or second capitalship is basically required so you can heal your starbase with repair cloud.
Also an amusing side effect of this strategy against human players is that it encourages them to build light frigates to fend off your Starbase, which means there fleet will be more vulnerable to your LRFs in the next phase of the game.
Finally, are there significant differences in the playstyle of the two Vasari subfactions? I know that the loyalists can destroy planets and become nomadic and the rebels can give allies buffs but neither of these advantages seem like they would come into play much in the average game.
You don't have to go Nomadic for the Vasari loyalist advantages to come into play. Having your capitalships generate credits and count as research labs alone is a great boon, since it lets you scrap some of your research labs for trade ports.
Also the two factions vary on their extra weapons research, so your late game fleet composition can be quite different. The Vasari Loyalists get rediculous (+70% damage) bonus to wave weapons, which makes their heavy cruisers (the only real ship that uses it) monsters. The Rebels on the other hand get additional bonuses to the already insanely powerful Vasari phase missiles, so they'll tend to go all out on units like bombers and assailants that have them.
