Just to add my $0.02 to the mix, I usually play Advent and illum spam is definately something I do a lot, so I'll offer up what my fleet is usually composed of, as well as a response to some of the counter-strategies already posted.
My fleet is usually composed of (just to give an idea)
120-200 Illuminators
30-50 Defense Vessles (anti-fighter/bomber)
30-40 Guardians
20-30 Subjugators
Halycolon Carriers (Building Fighters only)
Mothership(s)
Battleships, late game
The reason I've added my fleet composition to this post is that I realize that carriers are an obvious counter to the Illum spam, and I usually throw in many anti-fighter ships into my fleet to combat this, as well as building only fighters out of my carrier capital ship(s) as well. The guardians and subjugators keep my illums alive a bit longer, and the push from the guardians is almost priceless.
In smaller maps I've found that if I can get anywhere near the numbers I've listed above it's basically GG for anyone else I run into. Meaning if you cut the lower numbers in half, I can usually win with a fleet of that size.
On larger maps you're going to have to out produce me, which means getting a huge economy for TEC, or Returning Armada for Vasari. For this reason I usually go after Vasari opponents first because even in large games Returning Armada takes a LONG time to tech up and then afford to build. Once running you'll most likely be throwing ships at me since most of your ships are essentially free and the only thing you'll lose that cost you are capital ships, which can easily be afforded with the money you save on production. After awhile you'll whittle down my fleet and eventually defeat it, and that is when you know it is time to go on the offensive. Don't be afraid to lose an enguagement, but make sure that you can replace the ships you lose easily. Advent take awhile to rebuild an entire fleet, where the other races tend to turn over their fleets a few times in a game. Also, don't panic and spam cobalts just to fill out a fleet, illums cut through them like butter.
Some strategies to help this idea might include building phase jump inhibitors. If I go on the offensive you don't want me running away when you have me in your lap.
In response to the carrier strategy: I am very afraid of mass fighters/bombers, and if I see you spaming carriers I will increase the number of anti-fighter frigates to my fleet as well as guardians and halycolons to repell them away. A good advent will always have a scout on you and shift their fleet composition from pure illuminators to combat your carriers. The carriers are a response to illuminators, so when you mass carriers and get a decent size fleet to combat mine you need to strike first before I change my fleet composition to match yours. The advantage goes to you on larger maps since it will take time to move the new ships to my fleet, so strike before they get there. This is also a good time for you to molest my planets with a secondary force, if possible, since my production will be focused on re-balancing my main fleet than building another. Again, once you realize you have the upper hand go on the offense until I am completely gone. If you stop and let me rebuild you may end up facing an Advent with a counter to your fleet.
The secondary fleet strategy: While the idea that Advent don't like splitting their fleets is true, this doesn't seem like a hard-counter to the Illum spam. Most of the time while I am on the offensive I have an infant fleet brewing, usually composed of ships I plan on using to reinforce my main fleet with. Namely, more illuminators and, since this is later in the game, more powerful support ships like guardians. Not to mention I also will have some capital ships there who have not seen the light of battle yet. In short, your attack may seem like a good idea from your standpoint because I don't want to split my main fleet, but if you are careless and don't scout a little farther into my territory your secondary force could end up just being a means for me to level up some virgin capital ships a bit before I add them to my main fleet. Even worse, if the battle goes too far in my direction you may end up with the opposite. Instead of having me on the defensive on two fronts, you may end up on the defensive on two fronts.
One counter that I have had a hard time figuring out how to beat except with HCs is that of flak frigates and hokishos for TEC. Flak frigates are tough, and with their AOE attacks I have a difficult time Microing my fleet of illuminators around so that their attacks are not as effective. Because of the side beams on my illuminators I usually keep my fleet formation tight, so if I run into an enemy fleet i shoot as a small ball in an almost 360 degree arc around my fleet. Flak frigates can focus on this small area and tear my illuminators to shreds fairly quickly in large numbers. Just something else to think about.
As a side note: If I don't think my fleet is going to be able to stand up to yours I will build some culture buildings to slow down your advance. While this may not help my fleet at all it will slow your advance to help me get some more ships produced in the meantime, so make sure that even while you are winning a fight you continue to produce your ships to combat my fleet. Destroy those culture buildings!
I hope some of this helps those of you wishing to counter the illum spam know what one is thinking when you fight them. I know the tactic seems like an easy way to win, but understand that the people who are playing the build do not just play it because its all they know how to do. Whatever your counter to our fleet we will attempt to outfit our fleet to counter you. For me its a small game of chess. I make a strong basic move and look for your counter and then attempt to counter your counter.
