Advent culture isn't really any better than TEC or Vasari culture, it just becomes available with fewer labs.  Once maxed out, their +6% mitigation bonus from culture is very nice, but the amount of investment in the harmony tech tree necessary to make this a reality is impractical if what you're looking for is a military edge.

When you build a culture generator (Temple of Communion for Advent), culture will spread outwards from this planet to all connected planets.  The presence of any culture (even your allies) or enemy capital ships will repel it.  If a planet comes under your cultural sway, its loyalty will swing.  If the planet's owner is your ally, the loyalty goes up, if the planet owner is your enemy, it goes down.  If culture remains on an enemy planet for long enough, it will turn neutral (though this can take quite a while).  Planets under the effect of friendly culture will give a military bonus if you have the right techs researched.  For Advent, it will increase your maximum shield mitigation, for TEC your antimatter regeneration, and for Vasari it's your damage that gets boosted.  Allies gain the benefit of your culture.

The Halcyon gives Advent a powerful carrier on which to base their early-game fleet.  Capital ships are highly resistant to attacks, move faster than cruiser carriers, and do not have antimatter limitations.  With "Adept Drone Anima" to boost your squad count, this gives you a reliable early-game carrier.

Late-game, the boosting power of concentration aura from the Rapture gives Advent excellent damage output on their strike craft.  Not quite as good as maxed-out phase missile upgrades (by far the best weapons in the game) make Vasari bombers, but pretty awesome none-the-less.  You generally want to go heavy on bombers, since they will dish out a lot more firepower against frigates than fighters will, and better resist flaks.

Iconus Guardian + Progenitor Mothership; shield projection will spread out the damage your units deal, and shield regeneration will negate most of it.  This will take a lot of the brunt out of attackers.  Use the Halcyon with  TK push and guardians with repulsion to deflect incoming attackers and give you time to wear them out.

The primary benefit of Culture is economic, not military. Since the economic output of your worlds are limited by their loyalty, boosting that loyalty boosts their economic output accordingly. A few well-placed culture generators can give a 10% boost to your non-trade income in this manner for a relatively large empire, with the added benefit of preventing an enemy from colonizing unoccupied (or recently devastated) planets within your cultural influence.

Disciples should form the backbone of your fleet. They're cheap, tough for their cost and kick out a good amount of damage. Their steal/transfer antimatter ability is what fuels the late-game unstoppability. Back them up with a LOT of fighters, so as to swiftly destroy enemy long range frigates, the one true hard counter to your Disciples.

Disciples and fighters are the strongest counters you can muster against the Vasari phase missile ships that are your bane: Assailants, Sentinels, Transporters and Bombers. Disciples also will rob your opponent of Antimatter, starving their abilities and feeding your own.

Zeta,

as a long time advent player I will tell you about a combo that you will like.  This is rare to see in multiplayer but easily done in single player (mainly because you have to built the horrible advent battleship).

You have a progenitor capital ship use its offensive ability on a high hitpoint ship (preferably a capital ship).  The radiance battleship needs to be at lvl 6.  Then you use its cleansing brilliance ability on the ship you targeted with the progenitor while the progenitor is increasing the damage all ships in the enemy fleet take. Presto! You have a beautiful way to completely turn the tide of the battle. 

This takes some skill to use, and in multiplayer building a radiance as your first or even your second capital ship will likely cause you to lose.

Another one of my favorites is to lure an enemy into a "weak" planet and let them bomb it for a little bit.  Then you warp in a fleet that you have kept hidden from their scouts that has the guardians repulse ability, and that has at least equivalent firepower of the enemy fleet.  Then you focus down their capital ships (while kiting yours around the gravity well, retreating them if necessary).  Set up your guardians so that they prevent the enemy fleet from leaving the gravity well.  If done properly you can stop enemy reinforcements, kill all enemy capital ships, and destroy their entire fleet.  Watch the antimatter levels on those guardians though, they like to run out quick!

There is an upper limit for all factions, but it's pretty high.  As a human player, I've never actually hit this cap as any faction since mine-spamming like the AI does is an extremely bad approach to using mines.

I suspect that part of the reason why the Advent pack mines is because they use Aeria Drone Hosts as minelayers exclusively since Drone Hosts gained the ability.

So I am a newbie to SINS, just bought the full Trinity pack which has been alot of fun. Related to this topic, I am having trouble with the AI on the Advent Iconus Guardian. It never seems to activate Shield projection when I have it in a fleet. I am having to micro this every time. Do I need to disable auto attack on them so they only use their abilities? Any help would be much appreciated.

Perhaps you are thinking of repulsion?  The autocast is off by default for that ability...

In either case, both abilities are considered "channeling" which means the ship cannot do anything else while using the ability...so, if you are constantly giving the ship move orders then it will never be able to continue using its ability for more than about 2 seconds...

Good to hear you got it figured out  I would say sticking support ships on your caps is a wise idea...I generally play on normal speed so it's not as much of an issue for me, but on faster speeds I think most players tend to do as you have suggested...