'Speed' or velocity isn't really an issue, even today. We could easily build a probe and send it off in some direction at an incredibly high velocity if we wanted, by which I mean non-relativistic velocity. Gravity assist such as the orbits used by New Horizons is a gift that keeps on giving if you're prepared to go round and round again. The thing is as far as solar system science missions are concerned, you don't want to go too fast. New Horizons was already far too fast to go into orbit around Pluto. Much quicker and the fly-past wouldn't have garnered much useful data. There's just no demand for going really fast today because you've always got to slow down when you arrive where-ever you're going.
The feasibility of journeying to another star system has nothing to do with the distance. It's about how long it would take. Hundreds of years, thousands, maybe a lot longer. Do we have the technology to sustain life and the technology the life relies on? Can we put people to sleep for really long periods and then safely awaken them? Or do we build vast colony ships where generation after generation of people live and die knowing that it will be their distant descendants who colonise the new system. Nothing about such an undertaking is utterly implausible. Economic, social and political pressures will make it inevitable one day. People do pretty mad things to survive or in the hope of better life, you only have to look at the situation in the Med at the moment to see that.
When the first Homo Sapiens left Africa they weren't thinking about how unimaginably far the other side of the planet was or how they'd never get to see the Pacific Ocean (obviously because they had no idea what lay over the horizon, but you get my point). All they were thinking was "I hope we can find somewhere in that direction where the gazelle are a bit more abundant, the lions and bit less so and the neighbours aren't so noisy and won't keep stealing our food". Fast forward a hundred thousand years and the species has colonised the entire planet, literally one step at a time.
If (and it's the biggest if there is) we learn to live away from this place, either in space or on other planetary bodies then hopefully we'll make ourselves immune to the sort of catastrophes that could wipe us out while we are whole dependent on this single home. Then it's just a matter of time, and that's one thing there's plenty of, cosmologically speaking.