We've got 3 major hurdles to clear, technology-wise.
This is currently the big issue being talked about for future manned space missions. A lot of ideas are being thrown around, but no one has a totally great option yet. With slow travel of months to years in space, that's a lot of rads you could be soaking up. Maybe if we're lucky with all the push in material sciences, we might find some kind of nano material or something that works really well at radiation shielding without being too heavy.
Self-sustaining life support
Those months and years of travel without resupply means food, water, and oxygen are problems to deal with. Our attempts to create enclosed ecosystems have rarely lasted more than a few months before failing. I'm distantly optimistic we'll have a better shot at this with genetically modifying plants. (Since, we're probably going to have to do a lot of that for Earth anyway, in the next 100 years between topsoil depletion, desertification, rising water levels, and carbon reduction...)
The biggest hurdle is escaping gravity. We need to be able to get enough stuff up in space that can allow us to get to anything else, in order to farm those resources. The problem is that it's expensive in terms of money, manpower, and material resources.
We either need to find a way to synthetically make rocket fuel, cheaply, or find some physics-changing way to get to space. It needs to be cheap enough, and we need to have the whole launching process down, to where we can just do multiple launches one right after another, to get enough tonnage of gear, ships, people, resources, etc. up into space to start going other places.
So... assuming we manage to maintain our tech levels and not fall back under ecological collapse (which...is honestly optimistic given both scientists and military projections...), let's say 100 years to GET the science and the infrastructure to start getting up into space consistently with enough gear. We could get a moon base much sooner, but it will probably hold the same position that the ISS does now - good for research, but still needing a lot of regular re-supplies and not anywhere near self sufficient.
Maybe another 200-300 years to really colonize multiple other planets? Part of the problem is that collectively we're more likely to focus on working on one planet / space resource at a time. Each environment has such different problems to tackle, and it's easier to keep building on the lessons learned than to try to go take on completely new problems.
Colonizing space is absolutely NOT like colonizing other parts of Earth - instead of moving to a new land, or a continent, it's like digging a mine shaft and trying to live in it. You don't have food, running out of air is an issue, and the only things which you can get are valuable but don't necessarily keep you alive. (yes, we could mine ice, which gives us water, and we can crack it for oxygen and hydrogen, but how far are we gonna go for that? It also doesn't give you arable soil...)
Realistically, though, I'm guessing the next 200 years of human life are going to get really rough as our food and water ecosystems fall out of kilter and non-sustainable resources that push much of our technology run out. Ecologists predict this. The scientists working for big oil predict this. The scientists working for the US military predict this. That's going to put a massive crimp in us doing pretty much anything else when basic needs start to fail.