Well, the main radiation you’d have to worry about would be ionising radiation. The Van Allen belt would be a bad place to hang around for long, because it tends to trap and focus a lot of radiation in a small area. But if you’re in an enclosed metal capsule and travel quickly through it (as most astronauts and cosmonauts did), then you can get through it without anything drastic happening. . .in the short term, at least.
Out beyond that, you mainly have the “free flying” radiation. There’s always some, but the amount varies considerably from time to time and place to place. If you were to go out on a long spaceflight during a period of major solar flares, for example, then you’d be exposed to a lot more radiation than you would when the sun was being all quiet and peaceful. The sun tends to have a predictable cycle of such things, which lets planners plan ahead a bit. That doesn’t mean that the sun can’t go and do something surprising even during its “quiet” phase, of course.
Then there’s the radiation that comes flying through space from other suns, or even other galaxies. I’m not sure if there’s any sort of cycle with that. . .I wouldn’t expect there to be much of one.
So there’s always going to be a good bit of radiation when you’re in space, but you can time your flight to avoid much of the worst of it. And a well-designed spacecraft will shield you from much of it, too. As for the rest that does get to you. . .well, the human body can actually handle a fairly hefty dose of radiation without anything drastic happening in the short term. In the longer term there can be all sorts of health hazards. . .which is exactly what we’ve seen with many of the people who first went into space. Especially with the cosmonauts, it seems. The radiation-induced eye damage as has already been mentioned, for example.
I seem to recall that a lot of really high-energy particles came flying through space and left all sorts of microscopic tunnels through the spacecraft and people, and that microscopic analysis of some of their equipment revealed lots of these little holes. Obviously, that would be a problem over a long term of exposure. But for the length of time that these people were in space, it wouldn’t really do much.
As far as I’m aware, the moon itself isn’t any more radioactive than the Earth is. It gets hit by a lot more radiation from space, but the Earth and the moon are made of the same stuff. . .and the moon has a lot less of it all.