The basic Vasari early game strategy is quite simple: nanos and assailants.  This combo makes one of the most devastating early capital ship killers in the game.  Vasari will need to pressure enemy capitals to win a battle, since TEC will usually have the stronger economy (and therefor bigger fleet) and Advent has some jaw-dropping units at its disposal.  You'll want to use your scouts in order to capture neutrals and be aggressive about it.  A Vasari player who lets his enemies own their neutrals without a fight is wasting one of their faction's key advantages.

That said, I'm really in love with Vasari strike craft.  Their phase missile upgrades (the only attack upgrade in the game that affects both fighters and bombers) are absolutely phenomenal, and well worth the effort to get them on the field.  You have to be a hawk about flaks if you do this, but I find it's worth it, especially if the enemy refuses to get fighters of their own and you have free reign to bring out bombers in force.

Continuing to milk that subject, there are quite a few ideas on what you can do to better use you scouts as Vasari.

1] As you send them to scout, turn OFF the autocasting on detect mines. It REALLY helps them choose their next planet faster. With mine detecting ON, they tend to lag after every jump. Without it, they quickly select another planet to scout.

2] When you find an Asteroid Belt or any neutral extractors without rebels/pirates, order your scout right between the asteroids, turn off the scouting and leave the capture extractor ability on autocast.

3] If there are rebels/pirates in the gravwell, use shift key to order your scout to move all the way across the gravity well 10 times in a row. Pirate ships won't catch it, it won't get hurt. You have to remember about him though and once a while you have to manually stop that game of tag, order the scout to capture an extractor and once it's done - queue another 10 runs through the gravwell.

4] Always start with at least 3 scouts. You can get as many as 6 or more, even if you don't plan on using them in combat. But you NEED those neutral extractors and you NEED scuots to get them.