First off, in trinity Advent is by far the weakest race(and in every version of the game is the race that require shte most micro)- that said, if you're just playing against AI rather then going up against another player playing VAsari, this shouldn't really be an issue. anyway some general tips:
Early Expansion:
Grab either progen Mothership or Halcyon Carrier for your first capitalship. In the former case your first skill is Colonize, in the latter choose Extra Drone Anima. Start the game off loading either capitalship with fighters.
One of your biggest concerns to keep your expansion quick is for your actual mobile fleet to waste as little time in each gravity well as possible. This generally means if your capitalship or any light frigates ar epreset you only use them to kill the defending siege frigates, then colonize, then immediately begin building a turret, kill any long range frigate(the Javelis missile frigates), then move on tot he next gravity well. Don't waste time killing any cobalts or Gardas- the turret you build will handle those while your fleet moves on to colonizing the next planet.
Gravity wells with Heavy cruisers defending might need a bit more muscle then 1 turret to clean up though.
Make sure you give attack orders to your fighters- they are extremely effective at killing both siege frigates and long range frigates(the only 2 things you need to clear out before your fleet moves on)- but they have a bad habit of attacking other things if not given orders.
The most extreme case of saving time with turrets comes in the case of asteroids. All you need to capture an asteroid is a single colony frigate- no combat ships. Just send the colony frigate in and colonize and immediately begin building a turret as close to the asteroid as possible on the side with the defending siege frigate. The turret will finish building and kill the siege frigate before it kills your newly colonized space-rock.
Which brings me to my next point: make sure to expand in multiple directions at once whenever possible. If you start next to an asteroid and an ice planet, you can start by building a colony frigate and send the colony frigate to the asteroid and your capital ship to the ice planet. While the Capitalship is killing siege frigates at the ice planet you can use the above method to capture the asteroid on it's lonesome.
Later on you may send combat frigates to clear one well while your capitalship clears another.
Early on you should be very frugal with your resources. All of your money goes into upgrading the max population of the planets you capture, building extractors, and building more ships to speed expansion. Don't bother with upgrades until later. Make sure to scrap turrets once they finish cleaning up their well and it's a very common move to scrap your capitalship factory immediately after building your initial capitalship- you shouldn't be building any more anytime soon if you're building an economy and doing so frees up precious logistics. Try to avoid building too many logistic structures early- early on the only building you generally should build are extractors, a few labs to get key techs(ice/volcano colonizability, repair bays, and in some cases maybe culture buildings), or a new frigate factory closer to the front if you get into a war.
Note for the bigger planet types(Desert/Terran) a Prgen alone won't be enough- you'll need cobalts or scouts backing it up. The Halcyon however can handle these larger planet's alone as long as you keep it kiting the defenders while it's fighters kill any siege frigates or LRM frigate(as usual once these 2 types are dead the Halcyon jumps out and you brign a colony frigate).
Try to avoid getting ships killed when possible- colony frigates should at the very least survive to colonize 2-3 worlds, ideally more. If you see any neutral asteroid fields, make sure to send a colony frigate there to capture the neutrals. Once you get to T2 hostility and have repair bays, try to retreat combat frigates before they die to be repaired and sent back into the fray.
Building an economy:
The Advent has the weakest economy of the three factions, but the theory is more or less the same. Upgrade plane tpopulation early and often, build extractors, capture neutral extractors when possible, and eventually build culture and a trade route.
As for culture a single temple of communion at your homeworld generally will do.
Trade-routes however are where the real money is. At T3 harmony the Advent gets access to Trade posts- these are logistics structures that give credit income. Additionally they gain bonus income depending on how long your longest trade route is. A trade route is determined by the shortest continuous path of adjacent planets you have built trade posts between 2 planets. Note that you can actually shorten a trade route by building new trade posts that create a "shortcut" between the two endpoints of the route- so building trade ports at every single planet can actually hurt your income.
Trade routes are a pretty cimple mechanic that is rather hard to actually explain in a forum post though- a good way to get to understand how it works is just try building trade posts and mousing-over your credits at the top of your screen- this will highlight your current longest trade route on the phase lanes.
Defending your borders:
One mistake newer players often make is attempting to turtle too much. Static defenses by and large aren't very good- the game is designed in such a way that offense outscales defense. WHich is mostly to say don't spedn a ton of cash early on on turrets & hanger bays to defend your planets- that money would be much better spent actually just building more frigates.
That said, there are 2 very efficient static defenses: Repair Bays and frigate Factories. 2 Repair bays at a planet is all you usually need and makes a huge difference supporting a defending fleet. And nothing makes a planet harder to take then a frigate factory or two spitting out constant reinforcements.
That said, frigate factories use logistics, so you don't want to be building them all over the place. Once planet near the front should do. Also it's often better to have 2 frigate factories in one location then 1 factory each in 2 locations. Having factoriess concentrated means the enemy has trouble killing reinforcements faster then they arrive.
The Peneultimate defense is the Starbase. Strabases are extremely powerful, but also extremely expensive. Proper starbase placement is one of the hardest things to learn. you don't want to waste money building many of them- so it's all about anticipating where the opponent's attack will hit AND anticipating that you don't be able to deal with the enemy's attacking fleet without the starbase. This happens most often when you realize you are going to be hit by 2 enemies at once and can grab a choke point both have to go through. But again the biggest piece of advice I can give is don't over invest in these- if they aren't absolutely necessary you're better off dumping money into your economy or fleet.
Building a Fleet and Crushing your enemies:
First off I'd recommend learning the system of counters in the game. your canf ind the damage rations here:
https://forums.sinsofasolarempire.com/379562/page/1/#2677224
The most important early game counters are:
Longe Range Frigates counter Light Frigates, Fighters counter long range frigates, flak counter fighters, and light frigates counter flak. ALso Long Range frigates in particular are a lethal threat to capitalships/.
As the game progresses it will become increasingly important that Light frigates counter carriers & support frigates, bombers counter capitalships & structures, and fighters counter bombers.
Late game Bombers utterly wreck Capitalships & heavy Cruisers.
There are more such relationships but those are the most relevant. The biggest thing to remember is pay attention to your opponent- the best fleet is the one that counters his fleet.
That said, in trinity, massing LRF or carriers fielding bombers are the two "mainstream" strategies. Of these two strategies Massing carriers is generally the better choice for the Advent as they have concentration Aura, can repulse enemies away from their carriers, and the Illuminator is bad at focusfiring.
That said, if you do want to build a ton of Illuminators, make sure to bring flak(defense vessels) as well as flak counter the fighters which counter the illuminators.
Another mistake many new players make is overinvesting in capitalships or research. In general the best way to add raw firepower to your fleet is building more frigates. Capitalships are all about utility, and tend to be most potest when supporting a large fleet. Research mainly becomes cost effective only after you have built a large fleet.
Additionally Capitalships split exp- so each capitalship you build means all of your capitalships level up slower. And often due to the scaling of skills an antimatter limitations a single level 5 capitalship is better then 3 level 2 ones. This is particularly true of the Advent who have extremely squishy low level capitalships.
Try to avoid building more then 2 capitalship early on- and usually only 2 if you're planning to rush the enemy. Late game once you have a large fleet to support you'll likely build more though.
Advent Specific tips:
The advent has extremely cost-effective early game combat frigates, so one strength of the Advent always has been the "T0 rush". If you start very enar an enemy and go full-aggressive VAsari/TEC early game combat frigates will lose to advent onces, allowing for quick elimination of an enemy before they can even get started.
The Advent also has a very potent late game fleet. One thing to remember however is that the Advent Fleet is all about synergy. Most of their ship's skills are designed to be better suited to create synergy in alarge fleet rather then just making the ship it's self stronger. This means the Advent is all about creating a fleet which is stronger then the sum of it's parts- the converse is also true however, an advent fleet that's forced to split up loses more then the other races.
No mid-late games advent fleet is complete without Iconus Guardians. These things are the bread and butter of the Advent Military. they soak up huge amounts of damage your fleet would take, are extremely robust, and once you tech their upgrade gain Repulsion, an obscenely powerful skill when utilized properly. Combined with a Progenitor's shield Regen, Shield projection can render an Advent Fleet nigh-invincible.
However to unlock the guaridan you need to tech 3 shield techs, so don't do it too early or you can cripple your eco/military growth.
Don't bother with Adjucaters(bombers are vetter for killing structures) or Domina Subjugators.
Late game make sure to grab the Techs that improve shield mitigation, they are very strong.
It's particularly important when playing advent that you manually cast your ship abilities- by comparison tot he other races, very few Advent ships have sufficiently effective autocast AI(this is one of the reasons the Advent has more micro then the other races). The biggest example being Repulsion which can't be autocast at all(and is arguably the strongest tool in the whole race).
If you do build a large number of Illuminators, it is very important that you micro them in such a way that hey are always firing 2-3 of their guns at a time. An illuminator fleet that's only firing it's forward guns isn;t worth how much it costs to build.
Anyway, that's all that comes to mind for now. Hope it helps, and have fun 
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