Sorry, I've not had any problem taking on the Advent any more than any other race. I can not relate.

spend more time building your fleat and less time researching technologys so early in the game. infact, don't even bother with your capital ship until after you've capped your fleit and researched for more fleat. all you should really need early game is two civic centers and hangers, and make sure you get the planetary health upgrade on your front line planets. Look for choke points in the map too. For instance, say there's two points in the galaxy that blocks off half the galaxy, work on getting your fleats there and backtrack after you've established at those areas and make sure you keep large fleats within one jump of those areas at all times.

Why wait to get a capital ship until after you've established one planet and maxed your starting limit fleat capicity?

Five Frigates are 25 fleat supply = One Capital ship of 50 fleat supply. Even though the capital ships free, it still cost 1000 credits, which is the same cost of 2.5/5ths of two frigates. So for 50 fleat supply you'll get 10 ships that the enemy will have to individually target instead of what most people have, which is a maxed fleat capacity with 1 capital ship and four frigates. Skip the free capital ship for about 10 minutes and worry about using your military to gain economy. The first hour of the game basically decides the winner, researching isn't all to important until you can afford to research.