Never mind going to the moon, Mars or asteroids — let's send a probe out of the Solar System, to an Earth-like planet orbiting another star. That's the idea behind Project Icarus, dreamed up by the British Interplanetary Society and the Tau Zero Foundation. They'd like to send a probe traveling at 12% lightspeed, reaching another solar system such as Barnard's Star, six light years away, in about 50 years.
How the heck would that work? As the story goes, the spacecraft would first mine fuel from gas giant planets nearby such as Jupiter and Saturn. Then, the spacecraft's engine would fuse deuterium and helium-3 using searing heat from an on-board electron beam. The explosions from such a conflagration would happen 250 times per second, zipping the spacecraft to the stars. Once it arrives at its destination, it would fly by a planet, and send back as much data as it could gather in the short amount of time it would pass by that alien world.
Insert miracle here. Good luck creating a material strong enough to stand up to that kind of stress for a half a century. Maybe these guys could devise a plan to send a probe to Bernard's Star at a slower pace, perhaps over a few hundred years. But by the time it arrived at its destination, there might not be any scientists left here who care. On the other hand, during its journey, there could be dozens of faster, shiny new probes that overtake that obsolete spacecraft from the 21st century.