One strat I enjoy doing right now is to build 1 or 2 production tiles on my homeworld, (depending on bonus). For this I set my economy at max, with 100% social. Then I set it at 100% military with a colony ship to build, and depending on your military production bonus, you can sometimes get a colony ship every 2 days. Then I start grabbing any planet I can find, while scouting perspective colonies with my original colony ship and my flagship. The trick is to leave the new planets with only 1-2 maintenance buildings, no more. If you don't have any, just leave it empty and focus the planet on research. When you've several planets, switch your economy a bit to slow down ship production a bit and resaerch useful things. Recruiting centers are great for a new colony. Once you have your planets population grown and taxable, then you can start development and heavy research/ship building/what have you.

I do not know about production bonus. AI is supposed to get several bonuses on higher difficulty levels. However, I am fairly certain that AI knows the location of nearest habitable planet without scouting. Whether this should be called "cheating" is another matter. See the last post on this topic:
https://forums.galciv2.com/148553/page/3

I had no idea the AIs leased all their ships!

Seriously, I've been playing on challenging and every game I play I'm about even with the AI for the first couple years except for military strength and tech, but then my empire just explodes and I end up steamrolling over the whole galaxy.

It's just a matter of keeping them off my back until then. It's definitely time to bump the difficulty.

This is not a very fast rate of production and any AI at any reasonable level can easily match this in a number of ways.

I'll preface my comments by saying my experience is limited to DL and that I realize that there can be significant economic differences in DA and TA that affect the suggestions I make. However with that said I am sure that there are significant improvements that you can make to increase your rate of colony ship production.

When I'm going for a maximum quantity of colonies (which is not always the case), I have no trouble producing a colony ship every other turn on my home planet almost indefinitely. There are also times when I can produce a colony ship every turn for an appreciable period of time. Usually the limitation is being able to fill the ship with colonists as opposed to merely being able to build ships.

Basically it helps to have a 300% or 700% production bonus on your home planet but that's not absolutely essential. In DL it's also helpful to have your manu cap on your home planet as well. I know in DA the manu cap bonus has been reduced but it's not really essential anyway. Just pile on the factories until you have the necessary production. If it takes 10 factories then it takes ten factories. So be it.

But this is just the normal production way of building colony ships at a faster rate. You also have the rush buy/upgrade path either with or without lease buy options.

Generally the economics is the biggest limitation to any of these methods. In all cases I'll take a turn to research Sensors so that I can build a survey ship and another turn or two to research to Impulse II and then I can build a couple of reasonably fast cargo hull based survey ships to start searching for those money anomalies. That plus optimizing all your racial abilities and political party choices for economics can make a huge difference in how long you can maintain your colony rush.

Anyway once my survey ships are out looking for cash (usually by turn 4) I can then start my crash colonization effort. Certainly the faster engines helps out there as well. So basically you can go the route outlined above that is basically build up your production level to such that you can spit out a colony ship once every other turn. To this you can combine the idea that the first colony you build should quickly be turned into a second home planet to start sourcing colony ships as soon as possible. Generally this takes a number of turns as well bussing pop to the planet with another colony ship, but this can get you to the level of producing one colony ship per turn within 5 months.

However you can also go the rush buy/upgrade path and that can take a couple of different forms. I never buy anything on time but many good players do swear by it. Lease buy is certainly one way to (at least initially) cheaply fund buying a bunch of colony ships. This can be mitigated somewhat by producing for one turn then buying the remaining production to lower the cost.

I find the most cost effective way to "produce" colony ships is to rush buy the bare cargo hull (for about 400bc) and then next turn upgrade the cargo hull to a colony ship (usually less than 300bc). Your initial 5000bc (was this changed in TA?) plus a few money anomalies can go for quite some time doing this. In medium galaxies and below you can colonize pretty much the entire galaxy this way.

So essentially any player should be able to reach the every other turn rate of colony ship production in a number of ways. However, you need to be very spendthrift of your cash. Never buy factories on newly colonized planets expect if you're trying to setup a 2nd home planet. But other than that the best thing to build on any new planet is *nothing*. Wait for the planet to build up some population before allowing it to start building it's own factories. That initial colony building requires a huge maintenance. Don't be in a rush to make the planet cost more to support particularly when it will be some time before you can reasonably start producing anything on that planet anyway.