It is possible to establish a moon base and send a manned mission to Mars fairly cheaply. And all with (mostly) existing technology.

Check out Project Orion.

Nuclear pulse propulsion (or External Pulsed Plasma Propulsion, as it is termed in recent NASA documents) is a proposed method of spacecraft propulsion that uses nuclear explosions for thrust. It was briefly developed as Project Orion by ARPA. It was invented by Stanislaw Ulam in 1957, and is the invention of which he was most proud.

Calculations show that this form of rocket would combine both high thrust and a high specific impulse, a rarity in rocket design. Specific impulses from 2000 (easy, yet ten times chemical specific impulses) to 100,000 (requires specialized nuclear explosives and spacecraft design) are possible, with thrusts in the millions of tons.

This is possible because Orion uses nuclear power to make thrust without requiring the power to be held within a rocket chamber. Thus, very high temperatures, exhaust velocities and efficiencies are possible. Orion directs the thrust by using directional nuclear explosives, so it achieves reasonable efficiencies without a rocket bell.

In simple terms: Think of a cat with a string of firecrackers tied to its tail. Now imagine that with nukes. And a spaceship instead of a cat. That's Orion.

The main argument against this wonderful technology is an irrational fear of nuclear fallout. No really. The Linear No-Threshold radiation exposure model is only useful for environmentalist kooks. See here: http://web.ccr.jussieu.fr/curie.100/fulltext/jaworowski.html

And if you are not hot on using Orion to send men to other planets, it still has an incredibly practical use. It is the perfect technology for moving asteroids out of Earth-collision orbits when necessary.