Colonization and governance
For colonization you will need colony ships. As with all designs the custom designs can be better in one or other aspects.
The first question will be: How big a new colony should be? The stock colony ship founds a size 3 colony. One size of the colony requires size 15 colonizer module (absolute size, so on a size 5 colony ship it is a size 3 module) and the game will not allow you to put a module smaller than this (even though you seem to be able to decrease the size, it will just be a visual effect).
The easiest/cheapest to produce is a colony ship for a size one colony. However mid and late game, as the resource cost of the buildings increases this will slow down early colony build up remarkably (at level 1 it takes around 70 seconds to build the space port, at level 17 it takes about 45 minutes). In fact mid and late game you are better to just send in a true colony ship and remove the planetary capital (evacuating the colony in effect).
Above 3 the fresh colony will have production buildings (starting with a mine). If you do not want your colonies to have those than it is quite a waste. Above 14 no new building is added to the colony = absolute waste.
At colony size 3 having a city on fresh colonies is beneficial in terms of population and labor. However for a colony rush that population consumes food, goods and luxury. Having many size 3 colonies which you cannot offer to develop right then can negatively affect your economy.
A size 2 colony is a compromise. It is still cheaper than a bigger colony, it can be left undeveloped and its delay in building up versus a size 3 colony is constant throughout the game, around 72 seconds.
The second question is: Which planets to colonize?
For industry a bigger planet is better. A twice as big industrial block (consisting production buildings and cities) will produce four time the resources. However the AI governors are not really good at utilizing this. This is actually your chance against cheating AIs as they get a healthy resource bonus that you cannot match just by doing the same. The System window can help you by selecting visible systems, uncolonized planets and sorting them descending by free slots. Do not forget to click on select all. Then left click on the planet to show it on the minimap (to ensure there are no remnant ships around) and right click and set it for auto colonization - or send a colony ship manually.
I have to admit I have no idea how to select the planets for the second phase - mass colonization. I am aware of two ways:
1, Selecting all systems for auto colonization by zooming out and then either try to exclude remnant guarded planets or accept the losses.
2, Going through the list of visible systems and selecting the systems for auto colonization one by one.
Both of these are a pain to do.
The third question is: When to colonize?
There is no hard answer. Your colonization should not jeopardize the expansion of your economy. First I build a colony ship only when the metal production is overflowing (metal would be wasted). Once I have at least one production planet set up (preferably two) I usually switch off import on the homeworld and set the dry dock to continuously produce colony ships. Later an other production planet might join the efforts. When the galactic bank has more then 100K in metal all idle planets should build a colony ship - or more. This is the true rush.
As mentioned above the stock governors are not as good in specializing as a human player can be. As there is no vanilla way to edit governors, without modding the recommended way is to set the governor of new planets to renovate only and create and manually apply your specialized queues. You can create queues by adding buildings and ships to the queue of a planet then saving the queue assigning a name to it. The limitation is that you can only add as many buildings to the queue as there are open slots on the planet. Ringworlds with 100 slots can help with this.
The alternative is to edit the queue manually. As it can be done from within the game I do not see it as cheating. Usually the specialization includes a production (for bigger) and a research (for smaller) planets. It does not hurt to have an additional one for ringworlds.
After carefull assessment two colony ship designs got approved. Both were designed to establish a colony consisting the planetary capital and a space port. The bigger design (size 7.5) was for interstellar colonization with similar capabilities (albeit longer range) to the original proposal whereas the smaller one (size 6) was for intra system colonization starting from a foothold planet of the system. For the latter, speed was sacrificed for cost efficiency.
For the time being the Senate decided not to allow full independence to the colonies, but rather to have central governance with local governors being only responsible to build up following instructions and maintaining the planet.
Astronomers theorized that unstable stars might have the most ideal planets orbiting them. Rlac, within 64 AUs, was identified as a potential candidate. A specially improved high speed survey was built and launched towards it. Constructing the laboratories needed a lot of advanced components. It was decided not to consume the available stock so access to it has been limited but the advanced parts got mandated. Together with an academic campaign this helped to resolve the advanced part shortage so the mandate was put an end - and never ever used again.
The home system of the Terrakin empire was a miracle on its own. Apart from the homeworld it contained a true paradise (categorized as 26) and two smaller but still large planets (both 21). The beginning of the space era was surprisingly quiet. Apart from the orbital dry dock and the lone survey ship there was no space presence. It was not before the planetary storage went full of metal that the very first colony ship was constructed and launched towards the biggest planet of the home system. Soon all planets in the home system were colonized. The academic campaign was suspended to allow the first colonies to build up. The biggest planet was set as a production world - a template for many more planets to follow. The other two planets were individually instructed, one to provide additional goods, luxuries and food, the other to be the first science specialized planet of the empire. As Sociology was far from being a practical science, the first science planet was not made a template.