Heck, The Asteroids could be made into ships themselves, Just make very large hollow cylinders and they can fly among the stars gathering resources as needed pretty much as fas as the universe goes using not much more than the technology we already understand.
And that could happen after we have inspected every nook and cranny of our own solar system.
Oscillating between the null locations of the Lagrangian Points of the solar system, as they wind their clockwork way around, there is a natural path between the planets that has been called the Interplanetary Superhighway. It is the equivalent, by comparison to Hohmann Transfer Orbits, of a Hitchhiker's Guide to the Solar System.
We cam move cargo and colonies around, once we get into orbit somewhere. As Heinlein said, "Once you're in Earth orbit, you're halfway to anywhere."
Once we're well settled into the Solar System, we're halfway to the stars.
Humanity colonized all of Earth at a walking pace.
It was 1965 when a Liberal professor of chemistry at the university invited the public to an evening seminar, and the topic of the seminar was human interstellar space travel. Our teacher, a combat B-29 veteran of the 10th Air Force raids on Japan in WWII went to no little trouble and expense to take a couple of us to this seminar, because he knew it would interest us. Unfortunately, this Liberal professor spent the first ten minutes of the seminar ridiculing the very idea that humans could ever travel the vast distances of interstellar space and then changed the subject to discussions of Boron and other chemistry topics having no relation whatsoever to the topic of the seminar for which we traveled so far. The professor had the bad manners to refuse to hear from us or anyone else anything whatsoever about the multi-generation asteroid based interstellar spacecraft we knew were already becoming hypothetically possible with 20th-21st Century technologies. The teacher was highly amused when we answered his question about our impression of the university and its professor when we remarked how we had just seen an ill mannered and educated idiot in its natural habitat.