Where does the radiation in space come from and can we observe it? I have recently been reading that space travel is strongly influenced by "space radiation" and how it poses a threat to human space exploration.

Does this radiation originate from stars like our Sun, or is it an omnipresent — let's just call it — "force" in space (such as cosmic noise) that doesn't have any specific source?

Also, can an amateur astronomer visualize this radiation in some way so as to be able to observe it?
Cosmic rays consist of both electromagnetic radiation (i.e. photons) of different frequencies (radio waves, IR, light, UV light, x-rays, gamma rays), as well as charged particles (protons, electrons, maybe even ions of light elements), and other stuff like neutrinos. 

The vast majority of the radiation we encounter around earth will be from the sun, because it is so very close and basically a large radiating blob. Usually with isotropic (equally in all directions) radiating sources, the radiation intensity falls of with the square of the distance. That means radiation diminishes very, very fast. Go twice as far from the sun, and you only get a fourth of the radiation.

The EM radiation from UV and up (X-rays and gamma rays) is probably the most harmful. The earth's magnetic field shields us from these rays, but interplanetary travel will not have this benefit. X-rays and gamma rays may also come from supernovae and other stellar objects, which are far away, but will probably be much too faint to have an effect on astronauts. However, it can be picked up by sensitive specialized telescopes and satellites.

The charged particles may be a problem to spacecrafts and electronics onboard them, but can probably be dampened by shielding in the spacecraft, as to protect the astronauts. 

Neutrinos are I think of no concern, since they hardly interact with other matter.

As an amateur, you will have problems detecting UV and above. Mainly because we are mostly shielded from this kind of radiation by the magnetosphere and the atmosphere. 

You could detect particle radiation, by taking photos of northern lights, though... :)