There are two common models of space colonization people sometimes allude to, neither of which I think is particularly likely.
Model 1 (“normal colonization”) is that space colonization will look something like Earth colonization, e.g. the way the first humans to expand to the Polynesian islands. So your boat (rover/ship/probe) hops to one island (planet), you build up a civilization, and then you send your probes onwards to the next couple of nearby planets, maybe saving up a bunch of resources if you've colonized nearby star systems (eg your galaxy) and need to send a bigger ship to more distant stars. So it looks like either orderly civilizational growth or an evolutionary process.
I don't think this model is really likely because von Neumann probes will be really cheap relative to the carrying capacity of star systems. So I don't think the intuitive "slow waves of colonization" model makes a lot of sense on a galactic scale.
I don’t think my view here is particularly controversial. My impression is that while the first model is common in science fiction, nobody in the futurism/x-risk/etc field really believes it.
Model 2 (“mad dash”) is that you race ahead as soon as you reach relativistic speeds. So as soon as your science and industry has advanced enough for your probes to reach appreciable fractions of c, you start blasting out von Neumann probes to the far reaches of the affectable universe.
I think this model is more plausible, but still unlikely. A small temporal delay is worth it to develop more advanced spacefaring technology.
My guess is that even if all you care about is maximizing space colonization, it still makes sense to delay some time before you launch your first "serious" interstellar space probe, rather than do it as soon as possible[1].
Whether you can reach the furthest galaxies is determined by something like[2]:
total time to reach a galaxy = delay + distance/speed
So you want to delay and keep researching until the marginal spe