As far as radiation, the biggest thing that NASA is worried about seems to be as follows:
While fusion in the core does produce gamma rays, these high-energy photons are absorbed and re-emitted many, many times at lower frequencies in the solar plasma. However, gamma rays can enter the solar system from outside the system: these cosmic gamma rays do pose a hazard to the astronauts.
The Sun produces this form of radiation during solar flares. They can pose hazard to the astronauts if they get too intense, but so far there's been no indication of a solar storm severe enough to warrant emergency measures (which would probably mean evacuation of the station and a return to Earth for all astronauts).
Protons and Neutrons
These forms of radiation can be problematic, and are monitored on the ISS. (Electron radiation--called beta radiation--cannot penetrate the walls of the ISS.)
How much radiation is there? Enough that the ISS monitors those radiation levels constantly, but not enough for radiation poisoning or anything of the like. Astronauts are at a higher risk for cancer, cataracts, and cellular damage, but usually they don't experience enough for acute radiation-related health issues.
All information from http://www.nasa.gov/mission_pages/station/research/experiments/1043.html and http://www.universetoday.com/60065/radiation-from-the-sun/.