Figuring out that RAPID expansion and acquisition of materials seems to be THE priority in this game
You got it. This is your top priority; the first 30 minutes of the game are by far the most important. Usually the majority of the map is colonized by this point, so you have to make sure you get to any important planets within the first 30 minutes.
I was able to take over 5 planets on a 14 planet map before the AI effectively blocked my advances and I had to stop where I was and rebuild my fleet
You really need to have at least half the solar system under your command to have a fair shot at winning. You can scrape by with a 6:8 planet ratio on a map like this, but with a 5:9 ratio you're in for serious pain. The longer you wait, the bigger his advantage grows. The moral is that if you have fewer planets than the enemy, you need to be on the offensive.
The problem you have is that the map stabilized with his empire nearly twice as large as your's. The game then went on for hours, and the longer a game draws out, the more advantaged the bigger empire is. Early on, the big empire your opponent had wasn't a huge advantage. But after two or three hours, it became a massive advantage.
As an aside, five planets should take you no more than 15 minutes to acquire.
However, the computer was somehow able to create 1650 ships and just swarmed me after I tried taking over his nearest planet with the Deliverance weapon.
After 3.5 hours you can have pretty much anything. A deliverance was probably a poor use of money here, and making your fleet bigger to compete would have been a better expenditure. Moreover, a 5-planet empire is not big enough to justify more than 3 civic labs or more than 5 military labs. There are a couple of situations where you might add an extra lab to get a tech you want, but for such a small empire high-level techs are just too expensive and don't pay off well enough. Aside from unlocking important structures and units, you shouldn't be doing very much research at all with a tiny empire like that.
No matter what I do, the computer just takes over a few more planets at an amazing pace and creates UNGODLY amounts of ships and runs me over.
Well, there's a lot of room for improvement. I posted an Entrenchment replay a while ago for a player who was having similar problems to you. In that game, I colonized seven planets in the time it took the unfair AI to colonize two. You don't need to be that fast, particularly for the lower difficulty AI's, but it underscores the point that the AI isn't that amazing at all, and once you learn a few tricks you'll be running circles around it.
Having a significant bigger empire means that in the long run you're going to win. If your enemy has a bigger empire than you, you need to be attacking to try to change that. If the enemy maintains a substantially bigger empire than you for 30 minutes or so, you're probably going to lose.
With Advent, you should probably lead with either the Halcyon carrier or the Progenitor mothership capital ships. The Halcyon is a more aggressive choice, perfect for early battles, and the mothership a more defensive choice for nurturing an empire and supporting a larger mid-game fleet. In either case, these two capital ships work great together and it's often a good idea to have both. If you choose mothership first, get the carrier second, and if you chose carrier first, get the mothership second.
Your top priority is to scout. Find out the lay of the solar system and plan out your colonization route. You don't need to send all your units in one direction, and often sending multiple groups to colonize different planets is effective. Planets with 5 or fewer militia can be suppressed with frigates alone (6-8 disciples will do the trick). Send your capital ship to deal with the better defended planets. Sometimes planets with particularly heavy militia aren't worth your time. You can skip them (easy with a mothership, since it can colonize on its own and you don't need to sneak a colony frigate past) and colonize them later when time and opportunity allows.
Maintain a strong pace, and try to keep in mind when you will meet your opponent. You should build a frigate factory near the front line and start pumping units before a fight breaks out. Repair platforms near the front lines are a good idea, both as defenses and a place to fall back to if you need to retreat. The illuminator vessels is a great staple unit for your fleet, and combining it with the Iconus Guardian (you'll need a progenitor with the shield restore ability to use them effectively) for defense you can have a very deadly fleet indeed.