Early game, to advance quickly use the coloniser cap to grab as many worlds as you can and use corvettes in large numbers. Getting a second or third capital ship early on is not recommended since corvettes counter capital ships. Spend enough planetary upgrades to prevent underdevelopment costs from killing your economy.
2. To which tech I should rush with this faction?
In the civilian tech tree, get ice and volcanic colonisation techs if you have those planets nearby and get trade ports. Build one trade port in a grav well and then another trade port in an adjacent one in the longest possible unbroken chain. A[part from culture buildings there is not much that you will unlcok from the civilian tree.
In the military tree, corvettes are a must at first and then get carriers, defense vessels and/or illuminators depending on what your opponent is building. The basic shield and hull upgrades are handy once you start getting more ships. Guardians can repulse enemy ships out of range forming an effective wall.
I like getting the Resurrection, Reanimation techs and Protection of the Unity against the AI which are fairly good techs but not vital.
How is Advent any weaker in the early game compared to Vasari and TEC?
Early game, illuminators are not very good, their carriers (called drone hosts) are harder to get (although more cost effective than other races) and Advent have no repair ship. Their corvettes are adequate though, rely on those early game.
What makes them better at the start and what swings them back the other way later?
Advent gets stronger mid to late game thanks to their powerful Titan and their fleet synergy i.e. different ships work together to give a greater effect.
As a second capital ship after the Mothership, I strongly recommend the Discord Battleship. Use Fracture to debuff enemy armour, than use Malice to amplify damage and use the Eradica Titan's Chastic Burst to cause further damage. The Discord can use Lethargy as an even more powerful debuff against Titans. Very powerful combo. The Mothership should also invest points in the Shield Regen ability to help your fleet survive damage. This effect can be further increased by using a a guardian or two for their shield bubble.
Advent bomber spam is arguably the most cost effective in the game with Vasari a very close second. Having lots of carriers on the safe end of a grav well with a Titan, a few key capital ships in battle is a good strategy.
Late game, the Advent Rebels have access to the incredibly powerful Wail of the Sacrificed. This ability allows you to sacrifice a planet and its population causing damage to all enemy units and structures which are 1 phase jump away. However, the damage scales with population so only use it on well upgraded terran planets on your border. Be aware that you lose the planet but it can wipe out enemy fleets. This makes it difficult for the enemy to attack but it also makes defense difficult for them. Since population replenishes so slowly, use it wisely.