While I love MOO2, I'm remembering what was often my biggest issue in the game: adding new colonies. It always seems like I'm surrounded by terrible systems, with very few habitable locations.
I know I can somewhat correct this by taking advantage of freighters to have a few farming-heavy worlds feed all the more desolate colonies, but otherwise it seems like my only recourse is to crawl up the tech tree to things like Radiation Shield and Terraforming. Or play as Silicoids. ;)
Any suggestions on how to populate the galaxy?

I've tended to play as Psilons or similar, where the big research advantage means you can get terraforming and artificial planets pretty quickly.
A few research trades to keep your neighbours happy while you build your empire, then build a massive fleet and destroy them all.

Outposts mean that you can extend your effective empire radius (they also add a free Marine Baracks if you later colonize the planet).
Also, just grab the good (Abundant or better and Large or better) Barren and Radiated planets you find before your enemies do. Once you do have Terraforming and PRS, those planets turn into great colonies.
Also, there's a leader (Chung iirc) who gives you Terraforming for free if you hire him. If you don't get him, he'll likely be with one of your opponents, try spies or tech trading if that happens.
Also, focusing on research, production and food/population techs instead of spreading research out can help speed up access to Terraforming and PRS.
Anyway, combining Outpost creeping, grabbing good Barren/Radiated and research focus should help things out.
Or you could try to play in an Organic Rich universe, though that iirc cuts down on the amount of Rich and Ultra Rich planets.

MOO2 allows you to customise whether your universe tends to have mineral rich vs. organically rich planets. Don't forget that different star colours will hint at whether a system is likely to have a higher mineral count (and therefore productivity rating). I prefer to play in a mineral rich universe because I can always terraform my colonies later, however it IS easier to expand in the early game if you play in a universe with more lush planets.
This is the way I like to play.
Be Creative
I also prefer to create my own races and almost always choose to be Creative (my latest favourite racial combination is Low-G + Creative + Democracy). In my opinion, being Creative is a huge boon to your civilisation because it allows you to easily access all technologies that will help your colonies thrive. For example, a Creative race can access both biospheres and hydroponic farm technology right from the start. I find that technology will help you overcome any racial weaknesses, which is probably why Creative costs 8 points when creating a custom race.
Productivity
Any bonuses to your productivity are great. Research productivity boosts like automated factories quickly. The less time your people have to spend building, the more time they can spend researching for your future! Also, the more people you have, the faster you will do everything. Colonise larger planets, try playing as a Subterranean race (which doubles your population per planet) and research, beg or steal production technologies.
I think that next I'd like to figure out a Creative + Telepathic race...

Are you starting on pre-warp, normal start, or what?
For a pre-warp start, I usually start out on my homeworld using one or two colonist (depends on how productive the race is, I just want 5 or 6 production) to build something big (to save up production points without pollution waste); a colony base fits the bill well. With the other colonists (usually 3, without racial bonuses) I put towards research.
I research research labs first, which once i get them can be built in one turn with saved up production. I keep producing the colony base, and I research freighters. Once freighters are done, I research to Automated Factories. I build 1 freighter and then return to the colony base or other big item. At this point I probably have enough colonists at the home system that I can move another colonist to production.
This should take some time, so if I was building a colony base, I should have a new colony built. Once I have the colony, I immediately move an extra colonist there and make sure both are workers (food can be shipped from the homeworld). I want two colonists there, because it increases population growlth immensely. I put the workers to building marine barracks or straight to research lab (if the race doesn't need the barracks for morale). I occasionally just buy the research lab straight away, if i have the money and I think I need the extra research.
Generally, I stop colonizing the home system with just the first colony, unless there are other really desirable planets in the system. I would rather get c0olony ships and scouts made at this point.
In my home system, whether or not i made a colony base, I continue to build the big item until automated factories. I instantly buy those, then go back to saving up production. Then I research extended fuel cells (and whatever the other techs in chemistry are). Once that is done, I make a scout ship instantly and send it off to explore, then go back to the big item. Now it is time to research the colony ship.
Once the colony ship is researched, it can probably be built on the homeworld almost immediately, due to saved up production. As soon as I can, I buy the colony ship. I then move research to go for Pollution Processors. Once the colony ship is produced, I should have a good idea where to go with it because of the scouting. I then move to produce another freighter fleet.
Once the new colony is made, I do exactly the same thing as I did with the colony in my home system. I ship an extra colonist from my homeworld, and set both workers to building. I build automated factories first (buying them after they get to a reasonable price).
After that, it is just pumping out colony ships, scouting(replace the scout as needed), and researching until I meet another race. Once I meet the aliens, who they are and their relative power determines how i play the rest of the game.
I hope this helps. It took me a bit longer than i thought it would to type this out, so I hope you can figure out for yourself how the strategy could be changed for a start other than a pre-warp start. This is possibly not the best way to start, but it works for me. I am sure other people might have their own ideas about how it could be accomplished better.
I just played an Impossible, Huge, High Organic, Pre-Warp galaxy, and it was very easy with this type of race.
Race stats were:
Creative (8)
Telepathic (6)
Subterranean (6)
Repulsive (-6)
-10 Ship Defense (-2)
-10 Ground Combat (-2)
Replace Subterranean with Aquatic (5) and Large Homeworld (1) for a much faster start.
The game I am playing now is very interesting. I decided to make a Space Slut race (hey, the Elerian race makes it too easy). It was made insanely easy because my home system had a large Tundra (abundant) to colonize. You'll see why with the stats of the race:
Siren Race:
Subterranean (6)
Aquatic (5)
Large Homeworld (1)
Rich Homeworld (2)
+20 Spying (6)
Uncreative (-4)
-10 Ship Defense (-2)
-10 Ship Offense (-2)
-10 Ground Combat (-2)
Probably, I would have made a stronger race by replacing the homeworld bonuses with +1 productivity (3), but they are pretty strong regardless. Since I started with a size 28 and a size 30 planet in my home system (subterranean and aquatic FTW), my population was pretty insane early on. Come to think of it, this start would also solve a lot of the OP's problems, as long as he is okay with having to spy for required tech (uncreative is a bitch).

Been playing a bit with a friend. The GOG version of MOO2 is set up for network play; you just have to use the client/host game starters in the Network folder under the start menu shortcuts.
I've personally been finding that really big expansionism is pretty tricky. Creative isn't going to do it by itself-we played a game with stock races, and my Trilarians were significantly out-researching his Psilons.
Good races/setups for expansion:
Sakkra have Subterranean (much higher population caps) and cut a third off the price of a colony ship. They can expand like mad; but their Feudal government really gimps their research.
Silicoids need Cloning Centers to really get going because of their growth rate penalty, but because they need no food, ignore all pollution, and consider ALL planets to be Terran-class for purposes of population caps, they can settle in on even a Toxic-class planet and start putting in vacation homes.
Trilarians have Aquatic, which is very situational but extremely powerful in the right setting. In a Minerals-rich galaxy, they're going to be handicapped, but in an Organic-rich galaxy, they'll have free reign. They essentially treat most "habitable" world types as being much better than they really are; Tundras and Swamps get treated as Terran-class, and Oceans and Terran get treated as Gaea-class.
I don't remember if they treat the planet as Gaea-class in all respects; if so, it would pretty much make Gaea Transformation utterly useless for the Trilarians, since you can just Terraform up to Terran.
Klackons, once they get some population in place, can expand at an absurd rate due to their production bonuses. To put things in perspective: One time, in a game I'd already won, I got production (on Orion, true, but still, the point stands) up past 1000 per turn. I wound up producing a ridiculous fleet and attacking Antares with about fifty times the necessary force, all the while thinking "Hey, squidface. It's real funny now, isn't it? Puny race will be destroyed and all that, right? Real funny."

Any suggestions on how to populate the galaxy?

I have recently had the opportunity to play MOO2 for several months strait... to the point that the friend that introduced MOO 2 to me almost a decade ago suggested that I stop and play something else LOL but my expansion strategy is the following:

Lithovore +10
Tolerant +10
Feudal -4
Ground Combat -2
Ship Defense -2
Ship Attack -2

I create a game with galaxy size Huge, galaxy age average, players 8 (myself +7 AI) tech level Pre Warp. The key is being first to capture Orion for the massive tech jump. O and don’t bother with the Force Fields Tech tree it won’t be needed. And play on at least hard for some challenge!! Feel free to comment... I always keep an open mind. I always play this strategy FTW.