Well we have built "ships" capable of leaving the solar system -- we did that in the 1970s. Artificial gravity from a large spinning wheel it likely before we build generational ships.
I'd suggest the first test ones could be in place in the next 10 years (tether two dragon capsules together with a 4km long steel cable and a rotational velocity of 0.05 radians/second. The cable would need to be about 20 tons and be about 3cm across, that should provide about 1g in each capsule assuming that the dragons' structure can cope with it's mass being suspended at 1g from something near the docking port. There should be negligible difference in gravity between head and feet (about 5mm/s² when standing up)
That's a long way from a 4km wide rotating space-station, but it proves the point.
The problem with slower-than-light travel and intersteller travel is that you end up with generational ships, and you lose that planet bound culture. Once we can build a self sustaining colony ship, then we can build thousands.
But even at 5% of light speed that's 100 years between stars. Beyond the nearest dozen stars even radio waves will take 20+ years for a round trip.
When Europeans began to populate the Americas there was constant communication back and forward, even in the deep interior it would only be weeks for news to travel, not years. The environment people were in was basically the same, the timeframes were very human, and people traveled back and forth a lot.
Once the expansion wave starts pushing out across the galaxy (which is likely inevitable after the first few hundred colony ships have left) it won't stop, but in tens of millennia as the bow of the wave is 500 light years away from Earth, those that headed "north" and those that headed "south" will likely be a different species, let alone culture.
Humanities descendants may populate the galaxy, but it won't be a single civilization, not without FTL, and FTL means causality breaking
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