The basic approach is the same between factions, and it's the finer details that vary. Your biggest difference is capital ships and support cruisers, and then there's some variation in combat frigates and economics. Vasari have the most unique weapons and abilities, things the other factions simply have no equivalent to. This makes them very deadly, but also the most difficult to learn and play effectively (though once mastered they are the most powerful faction by a longshot, so you a lot of Vasari online).
Anyways, your top priority at the start of the game is to scout. You need to find out the layout of the map, where the resources lie, and most importantly where your opponents are. In close second place is to begin your colonization efforts. In the early minutes of the game, you'll lay the foundations of your empire, and that means spending your starting cash quicky and effectively to get yourself running.
The top consideration for your starting build order is where the nearest enemy is. As a rule of thumb, if they're 4 or fewer jumps away, you want a military opener, and if they're 6 or more jumps away you want an economic opener. This is a generalization; if there's a big desert planet with a bunch of powerful militia between you, this will probably prevent a rush, whereas a string of asteroids is like a chain of breadcrumbs leading both of you to an early conflict. You need to scout quickly so you can make a decision quickly.
Your first order of business is to pick capital ships, and most of the time you don't have the luxury of knowing whether you're in a military or economic situation yet. If you already know the relative size of the map, you can make a (pretty accurate) educated guess, but the problem is you just can't wait until you know for sure; you must pick immediately. The carrier capital ships are your top pick for military situations, while the colony capital ships are generally your best economic bet. For Vasari, most people go with the Skirantra all the time these days, but this is because its "scramble bombers" special ability got over-buffed last patch and the Jarrasul (your colony capital ship) is itself overdue for a buff.
Your second priority is select your labs. Generally people build two civic labs or two military labs at the start of the game. Vasari has the option of demolishing their capital shipyard after their first capital ship is done and building two civics and one military. While nothing stops the other factions from doing this, one military lab is completely insufficient for the needs of TEC or Advent, while Vasari can get by with it just fine. The purpose of civic labs is to unlock colonization of ice and volcanic planets, while the purpose of military labs is to unlock superior unit types. This early in the game you generally avoid other types of tech, instead focusing on making a bigger military and a bigger empire to get stronger.
For colonization, you should have a small group dedicated to pick off little asteroids that are out of your way. Save your capital ship for the big and important targets, and use frigates to take out the little ones. You should make a beeline for any strategically important planets while building up a military. For Vasari, your top military unit in the early game is the assailant. The repair platform is also a must-have, since you can retreat your capital ship if it's damage and restore it to full fighting order in almost no time flat (I've retreated capital ships, repaired them, and gotten them back to fighting in the same battle. That's game-changing, and it's something you'll do on a fairly regular basis).
The biggest problem I find from beginners is that they spend too much cash on technology and structures, and too little time and effort colonizing planets and building a military. In the first 30 minutes of the game, this is where 95% of your resources should be going: colonization and military units.
