NASA had a proposed "Interstellar Probe" mission that was to use a 200 meter diameter solar sail to travel 200 AU in 15 years (the heliosheath, or edge of the solar system, is about 100 AU from the Sun). It was thought they could keep contact until 400 AUs distance. Link (go to the Interstellar Probe Report link there):
This mission, as well as a proposed follow up mission (I think called Interstellar Probe 2) that would have had a bigger sail, gone twice as fast, and reached around 1000 AU, were both shelved quite a while ago. (There was a single NASA page on that second mission a long time ago, but I cannot find it now.)
You could use probes like these to sort of act as galactic weather probes, testing the interstellar space that our solar system would encounter in coming decades/centuries and seeing how that "interstellar weather" affects the Sun the the Earth's environment as we pass through it.
A theoretical improvement on a solar sail would be a "light sail"--you could set up satellites with powerful laser systems orbiting the Sun and use that focused and powerful light to push sails much faster. Some folks have hypothesized reaching 10% to 30% the speed of light using techniques like this if the lasers were powerful enough and coordinated enough. Assuming success, you could possibly send an unmanned probe to Alpha Centauri in a matter of decades. In this instance, you could theoretically use a solar sail to use the solar wind/light of the Alpha Centauri system to slow down once you started nearing it (although that could potentially add more travel time).
Things like these could be a relatively fast, cheap, and safe (as compared to nuclear) way to explore our external solar system (Kuiper Belt) and nearby interstellar space (Oort Cloud), and get a good handle on how our surrounding interstellar space affects our solar system. Very interesting stuff. I hope more of it happens!