Ah, close enounters. Lovely map.
This map has two asteroid belts between the players. These can have between 0 and 3 extractors each. If there are 3 or more extractors in total, they are a massive asset that must be fought over. Allowing your opponent to control these is unacceptable. Your first goal at the start of the game is to scout them and uncover just how many extractors there are. Your second goal is to send units and colony frigates (scouts if you're Vasari) to capture these. Allowing your opponent even a short period of time to control them is completely unacceptable. Send a few light frigates to capture your asteroid; don't waste your capital ship's time there.
9 times out of 10, you shouldn't build civic labs on this map. The ice planets may be tempting, but they are difficult to defend and chances are your enemy will kill you if he spends that money on military instead. Due to the close proximity of players you should be spending close to all your money on military and military research.
For units, you need to use scouts in combat. The scout is very effective at killing other scouts (Vasari will be using them to capture those neutral extractors) and long range frigates. Because he's after those extractors and long range frigates are the best damage dealers his faction has, chances are he'll be building lots of both. To counter these units, he will use capital ships, light frigates, flaks or carriers. If he uses flaks or carriers, counter with light frigates. If he uses capital ships or light frigates, counter with long range frigates.
A good capital ship to play with here is the Sova as TEC. Its fighters and missile batteries will give you an early-game edge in combat. It will be difficult for the Vasari player to counter it. If he's using the Skirantra, use fighters to kill his own strike craft. If he's using the Kortul, switch to bombers and pelt his capital ship (it has no fighters to defend itself). If he's using the Jerrasul, stay away from it at all costs (nano disassemblers is a killer) and use a combination of fighters and bombers. As Advent, you can use the same trick with the Halcyon, but you won't have missile batteries so you'll need more units to back you up (fortunately, Advent has very strong scouts and light frigates)