Our galaxy is a big place, but that doesn't mean we don't have any neighbors. There are a handful of stars close enough to Earth to be potential targets for exploration within our lifetimes. The hope was that one of the closest of these, Barnard's Star, just might have a habitable planet in orbit, but new observations show no signs of anything at all.
In order to send a spacecraft to another star in an amount of time that would strike the average person as reasonable — like a generation — we're going to need new ways of propelling the aforementioned spacecraft that would strike the average person as nuts, like using solar-powered lasers to coax antimatter fuel out of the quantum vacuum of space.
Voyager 1 departing our solar system — lest it smash into an asteroid or run afoul of a black hole, say — is seen as an eventuality, though we've waited a while. Now, NASA's Jet Propulsion Laboratory, based on data sent by the craft, is saying the explorer could become the first manmade interstellar object very soon.
Helium-3 is a trendy new possibility for clean and efficient fusion power, but most of the Earth's supply has long since floated away like a party balloon. To get enough helium-3 to power an interstellar spacecraft (and the rest of the planet for the forseeable future), Project Icarus wants to send floating robotic gas mines to Uranus.