Colonizing the galaxy by slow boating reality check
Humans like to explore and seem to have an almost instinctual need to expand. After spreading throughout the solar system and even into the Oort Cloud they decided that humanity should follow the robot probes out into the galaxy.
Without FTL, but having discovered a form of artificial gravity and having experience creating space habitats, they create a number of generation ships. These massive structures are initially given a population and crew of 30,000, with room to expand to 50,000, and each one has all the tools and manufacturing capabilities to make more generation ships and space habitats, along with the ability to terraform a planet.
Strapping on a few large meteors with ice, minerals and other things they may need in an emergency, these ships slowly make their way to the nearest solar system. Slow being a little less than half the speed of light, thanks to getting a very large boost as they start their journey.
Once there, they get to work making comfortable habitats for the now increased and cramped population using the resources of the system. 199
They spend several decades there, creating a working system of habitats and making any repairs that are needed on the generation ships. After a century or two the generation ships, possibly a few new ones as well, get crewed by people who want to travel and move onto the next solar system to do the same thing all over again. Eventually the fleet splits into two and each one does the same thing, eventually splitting again, and again and again.
If they find a planet that looks like it can be terraformed, a planet with no life, or only the most basic of bacteria which is wiped out, they get to work making the planet livable for the people who want to have a sky over their head. Any planets with multicellular life is carefully studied by probes, but left otherwise alone because the risk of contamination, allergies, etc, are too great for the ships, and terraforming them will destroy the ecosystem.
Is this a realistic way to colonize and explore the universe? 197