it would be great to travel one day around space? how can one do it cheaply without fuel?

as a child i was always under the impression that going to space, the moon etc would be very very expensive and also hard to achieve in terms of technology....

what if no fuel is needed or chemical rockets to propell a space ship forwards?

of course assuming you have reached LEO, or are taking off from the moon?
A.C Clarke did quote sufficiently advanced technology is indistinguishable from magic.

What about engines with no moving parts?

Solar energy converting xenon gas?

Electromagnetic radiation?

If the earths atmosphere can be tapped in terms of the ions etc why cant a vacuum be possible?

The cheapest way I know of is with a good telescope. A good Newtonian on a Dobsonian mount will cost you a few hundred dollars, a lot cheaper than rocket fuel!

Travel without fuel, and unlimited energy to run onboard systems are both highly desirable goals for anyone desiring to explore the stars. Some people have talked about tapping the vacuum for such energies, but most authors claim that vacuum energy is highly overrated. At this point and time Magic is the best answer. This can mean an answer doesn't exist, or it can mean (in the A.C Clarke quote) a sufficiently advanced technology is indistinguishable from magic.

"There's no such thing as a free lunch." The closest you can come to your ideal would be to "live off the land" in the solar system, finding fuel and using it as you go. The atmosphere of Mars will yield rocket fuel, as will the water ice on the moons of the outer planets and comets. I always liked the idea of perfecting a fusion reactor for space travel. That sounds like it would make interplanetary travel practical on a large scale. I wonder if it is possible to engineer, though.

No rocket or spaceship can work without some form of fuel, the closest i can think of would be "Solar/Laser Sails" because they can be powered by sun light.

http://en.wikipedia.org/wiki/Solar_sail

Or maybe a MagBeam would work as well but it would need fuel like argon or xenon to make it work.

http://en.wikipedia.org/wiki/Magnetized_beamed_plasma_propulsion

Electromagnetism. Tesla technology.

You cannot travel all around space, it would take you over a billion trillion years.

Due to conservation of momentum, there are two ways to get around. One is to throw rocks, and the other is to catch them. Throwing rocks has the issue that you need to bring the rocks with you. So let's look at some ways to catch them.

Solar sails. The light from the Sun has pressure. You can angle your sails to push you in the direction you want to go.

Solar sails with a laser beam from the ground. If you want to go where the Sun's light is dim, arrange to have someone beam a laser at you. The momentum you pick up comes from whatever sent the light.

Magnetic manipulation. A big loop of wire with a current can set up a magnetic field. You can use that to push off from another field - from the Earth, the Sun, Jupiter, or the Galaxy. There are several variations on this theme.

Bussard Ram jet. The idea was to use a huge magnetic scoop to gather the few atoms in space, cause them to fuse, and use the energy to push other atoms you collect to high speed, like a jet. We can't currently get fusion working on the ground. But perhaps you could have a fission reactor and just ionize the free atoms and accelerate them magnetically.

Any sufficiently advanced bug is indistinguishable from a feature.

The current episode of AstronomyCast talks about this. It's a good show.