The thing to remember about space exploration is that it is not like going from one place to another on the surface of the Earth. It is like climbing out of a very, very deep well (Earth), scooting across a barren surface, then diving down another deep well (the Moon or another planet). It takes a lot of energy to get up and out of those wells, and so there needs to be good payoff for it to be worthwhile. Unfortunately, the only thing at the bottom of the wells within reasonable distance of the Earth is... nothing. The moon is just a big hunk of rock. So is Mars. There isn't much point in expending all the energy to get there and back. If you want a barren landscape, the Mojave Desert is available. And it has an atmosphere.
Space is the most inhospitable environment for human beings imaginable. If we really want to start making things happen on other planets, the way to do it is to forget about manned space travel and use automated systems. A space probe doesn't care that it can't breathe in a vacuum and that it's -100 degrees C. Spend the money on engineers like me and not hot-shot flyboy astronauts.
The difference between space travel and the 15th century explorers was, as Derb points out, that there was a fairly big and immediate payoff to terrestrial exploration. Spices, gold, new and exotic plant and animal life, native girls... these things made the trip worthwhile. There aren't any native girls on the Moon.