Not sure if anyone needs this, or will find it helpful, but I'm gonna drop what I understand about the pop growth system. (Bugs/Exploits notwithstanding.)
For organic empires, your planets have an initial monthly growth rate equal to the (Colony's Population / 400), so you have a .25 pop/month growth rate on a new colony (100 pops / 400 = 0.25), while a colony of 1200 would be 3 pop/month, 2k 5/month, 2.8k 7/month, etc.
Then, a penalty is applied when your population is too large for the planet's capacity. This penalty is essentially meaningless for large colonies, since the initial growth rate is so high (and a cap is applied to this initial growth value anyway) that this doesn't matter. For NEW COLONIES however, it can negatively affect growth - so I just build a housing building (such as luxury apartments, etc.) and I find that gives a nice capacity boost and negates most of the penalty, if not all of it.
THEN, this initial growth value, with modifiers from capacity, is capped at 5 pop/month.
So, a planet with 2.8k and 40k pop get capped at the same value of 5/month (assuming you don't have massive housing shortages, inflicting a larger capacity penalty - just make sure everyone has a place to live.)
THEN, growth rate modifiers are applied - so at the capped growth of 5/month, your 20% growth bonus brings you to 6 pop/month. (This, to my knowledge, is not capped.)
THEN, add any pop assembly (with modifiers) to that, and you get your total monthly growth.
Now, to get your population up on new colonies, you have two options:
1) Resettle a base 'breeding stock' of pops to the new world.
2) Wait for pops to resettle on their own.
Option 1 might be available depending on your laws, and cheaper depending on other factors, but I've been playing Egalitarian, and just stick with option 2.
Each colony you own will, at a base, export up to 10 pops a month via automatic resettlement to other planets you control.
(This assumes the migrating pops are civilians, or are unemployed, and the target planet has jobs/housing... and that you haven't outlawed automatic resettlement. 
)
That means that at game start, with a relatively unmodified empire, your first colony should grow at a rate of roughly 10.25 pops per month on average (10 migrants per month from your homeworld, 5 of which would be new pops, the other 5 civilians, and then the 0.25 pop per month from your existing population). As the colony grows, it gets faster, of course. Just make sure to keep open jobs and plenty of housing.
And if my very lazy math is correct (and if I know how to use a graphing calculator) a fresh colony relying on migration should hit ~1800 pops 120 months after being colonized.
During that time, your homeworld would have had to send away a net 600 civilian pops (5x120).
The trick is that the second habitable world, if colonized, would split the emigration from your homeworld between both worlds, so I've been making sure to colonize my first planet quick, then wait for it to be seeded with a decent number of pops before plopping down a second world. (like maybe around the 800 mark or so.)
Eventually, once you have a few populated colonies, you'll find new planets populate VERY quickly due to the emigration from other worlds - all you need to do is build up the housing/jobs on the new planet, then stop building new jobs on your existing worlds. New pops will be born and flock to then new world.
Lastly - if your planets growth surpasses 10/planet, you wont be able to emigrate away fast enough - some techs that raise emigration "chance" buff this rate by the chance percentage. So Transit hubs give a plus 100% - that raises the emigration to an average of 20.
There's a bunch of probability under the hood that I can't be bothered to understand - some months no one will emigrate, and then your planet will recieve 80 pops the next month, but the average values seem to line up over the course of my games.
And, of course, if you outlaw migration, then you NEED to pay to resettle pops between worlds to get that growth rate boost - otherwise, correct, your colonies will never grow. The upshot is you can cheapen the cost with species traits/civics, and just immediately fill your first colony with 2k pops for great initial growth. Then 'save up' pops for your next colony, etc etc. (I think Corvee system, or it's equivalents, are pretty nice here to avoid the unity costs though... but I don't play Authoritarian usually, so I'm not sure.)
Anyway... uh, if anyone finds something super inaccurate in there, let me know. I think that's all been pretty correct, as far as I can tell, but I'm more than happy to learn a thing or two here. haha
(And yes, I agree, the game is horrible at conveying these details, but I actually really like this system - it just lacks in all transparency, so the only reason I know any of this is from reading forum posts after 4.0 came out.)
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