A critique of your replay:
Your first problem was in strategic choice of planet colonization. While your choices were acceptable from an economic standpoint (a tad too much backtracking), they left you militarily exposed. By 7:30, you'd scouted enough of the map to know that the desert planet Hercynia is a choke point between you and the enemy. You should have bee-lined for it with your Progenitor, or at very least the dead asteroid Tercinida. Simply capturing and holding Hercynia would likely been enough to win easily, cutting off the AI from attacking. The planet wouldn't be colonized by the AI for another 20 minutes; you had the opportunity.
Ignoring the dead asteroid Tercinida was a massive blunder. The AI was slamming you with units all over the place, and you could have plugged its advance with one colonization job, but didn't. Instead, you sent your Progenitor running in circles and let the AI capture this critical choke point planet, giving it free reign to attack you while leaving you defending multiple planets.
Later, you had problems with using "hold ground" behavior. This is very important at times, but in this situation it held you back. Your illuminators weren't able to use their side beams because they were too far away, and you missed the opportunity to chase down retreating enemies (who would be stumped for several seconds by phase jump inhibitors) by standing still.
You became much too tech-heavy. They were good tech choices, but you should have built up your fleet a bit more before pursuing them. Your capital ships were fine (can't go wrong with Halcyon and Progenitor) and you stomped the AI pretty well once you get some momentum.
Your micro/macro isn't terrible. You've got a ways to go, but you've certainly made some strides.
Was my opening okay? Besides smarter fleet control and more potent aggression, what other possibilities should I have taken into account if the enemy was a human player?
If the enemy was a human player, you'd have needed to scout a lot more voraciously and definitely bee-line towards the enemy. You can be rest assured, victory and defeat on this map hinged on controlling Tercinida (bonus points if you could manage Hercynia, but a competent human player wouldn't let you do that).
Honestly, though, your start location on this map was jaw-droppingly good. You had four asteroids just sprawled out in front of you, lots of planets in the back for later expansion, and nothing that would require civic investment initially. You could have easily gone for a pure-military rush and kicked the snot out of your enemy with a start location like this.
Would it have been worth it to invest even more in logistic slots/trade?
You were in a position where you could have played the long-term if you'd wanted to. Really, once you held Tercinida it was over anyways, and it was up to you as to how you wanted to finish it.
Was my fleet composition okay for this match? Was it right to later on build more LF to hunt down the AI's carriers? Or should I have gone for more illums?
Your fleet composition was alright. The AI's carriers were mixed in heavily with assailants, so you'd probably have been better off with more illuminators than disciples in this case. Disciples are great when the carriers are being flighty and you need to chase them down, but the AI tends to stand and fight, so you don't need to worry so much about that.
How should I be microing the illums to get the most bang for my buck?
Get them right in the enemy's face, guns blazing.
Any other suggestions on how to practice to prepare for human players? Should I just dive in and get stomped a few times until I learn the ropes?
You're probably at the point at which there's not much more you can learn from the AI. You can always practice more, but otherwise you're ready to jump into multiplayer.
Keep in mind that most people play with quick-start on in multiplayer (you start with a capital shipyard, all three extractors, and two scouts already built) and you'll have an easier time finding games in Diplomacy than Entrenchment.