I think it can go either way. expanding into the solar system is pretty much the only way we're going to continue the modern way of life because there just isn't enough resources here on earth. I don't think we'll "colonize" other planets but I think there will be mining outposts throughout the solar system, mostly on moons and astrioids, though.

but it seems incredibly unlikely to me that travel between solar systems will be both physically possible and feasible in terms of resources.

If we make the jump and start harvesting resources throughout the solar system, there will be more resources available to humanity than we will know what to do with. There are individual asteroids with more metals, especially precious metals, than are available on the entire earth, there are moons of gas giants that are basically blobs of hydrocarbons. The solar system is absolutely loaded with resources. If we survive the next couple centuries and start to mine asteroids and whatnot, there will be enough resources at our disposal to make interstellar travel feasible. Yeah, stars are extremely far away, so any sort of interaction between systems isn't really going to happen. But if we find earth-like planets in other systems, there's no reason we wouldn't be able to colonize them.

There's two ideas in how to reach those places- massive multi-generational ships and super fast ships that take advantage of potent power sources, likely tons of fissile material. The practically infinite amount of resources in our solar system would make such an endeavor possible, although it'd have to be publicly funded because there wouldn't really be any economic motivation to colonize other systems. For some perspective, it takes only one year of 1-g acceleration to approach near-light-speed. And thanks to time dilation, we could quite easily reach other star systems within a single life time. The closest stars would take ~5 years, stars further away ~10 years at 1-g acceleration. The wiki page on space travel using constant acceleration covers this somewhat well. Of course, we don't have the technology today for constant 1 g acceleration to be feasible for such an extended period, but the example adds perspective into how interstellar travel may become feasible in the future. thanks to time dilation, a round trip to even the Andromeda galaxy would take only 50 years on ship at 1-g (of course, millions of years would have gone by on earth in the time... so, yea, like I said, there's not going to be much interaction between colonies)