The AI is attacking in force before you even have your second planet? You playing against unfair or something, because typically in my experience the AI might send a few harassing cobalts but otherwise goes straight for those three terran planets. I'd recommend starting with a Marza if you're TEC; I don't think Akkan cuts it here. Evacuator and Mothership for Vasari and Advent should go without saying.
Your first goal on gaian crescent (or any map) is to immediatley take your nearby asteroid. Build a capital ship and send it over right away. You should now make preparations to take the desert planet. The trick here is that because of the map layout you don't want to invest too much money in your army. If you're TEC, you can take out the defenders with just a Marza. If you're Advent, purchase the second level of your mothership to get shield restore (always pick colonization first) and then you should be able to do it with only a few disciples. If you're Vasari, just get assailants since they'll be useful late game (but not too many).
You should be moving without stopping. Build your capital ship, take the asteroid immediately, then go straight towards the desert planet. Research repair bays while you're attacking the desert planet, and focus on killing the siege frigates first. Once the siege frigates are dead, use either your capital ship (if Advent or Vasari) to colonize the planet or quickly run a colony frigate in to take it over as TEC. While killing the remaining defenders, build a repair bay to bring your capital ship back to full health. Also build the labs necessary to research volcanic colonization, then immediately jump to the next planet to take it. Repeat the process (and bring in a few units if you like) to take the remaining two planets.
You can easily have your four starting planets locked down with this fast and low-cost maneuver. Now you want to start heading towards the terran planets and taking them out. Start building your real army at this point, particularly long range frigates, and take the terran planets. Because the AI produces lots of light frigates (and gets them killed...) to clear out those terran planets, your LRF-based fleet can mop it up nicely. Move fast enough and you can have all three terran planets under your control.
From that point on, I find the best way to win is to build two separate fleets for each side of the AI's empire. Use scouts to figure out where its fleet is, then attack them from the other direction. As soon as they arrive on the other side, run away and send in your other fleet. They'll spend all their time running back and forth while you kill their planets. Quite hilarious actually.
I actually played a wonderful game against the AI on that map earlier this week. I got hung up on their volcanic planet because it had high security protocol and they built a huge number of repair bays and hanger defenses there. Took me a while before I managed to break those defenders. I used the trick I described above to do it; two fleets playing cat and mouse from either side of his empire. Got a level 10 Marza and level 10 Kol out of it in the end, though, which is totally sweet 
