During the Apollo missions, there was the constant concern that the astronauts might be exposed to high levels of radiation. The Earth has a protective magnetic field and a thick atmosphere that helps shield the inhabitants from deadly solar flares from our sun and Galactic Cosmic Radiation, or GCR, comes from outside the solar system but primarily from within our Milky Way galaxy. (Highly energized radiation that comes from outside of our solar system is GCR)
This Highly energized radiation GCR bombards the surface of the Moon and satellites orbiting it. The Moon lacks a global magnetic field like that of Earth. As a result, the lunar surface are not shielded from SPE(solar particle events) that erupt from the surface of the Sun.
While on the surface of the Moon, Astronauts are partially protected by the Moon itself as it helps block some of that radiation. An astronaut on the surface of the moon is protected from the GCR environment in 2π directions by the lunar regolith(Surface). You can visualize this with a large exercise ball and placing a lego mini figure on one side(just use some tape to hold the figure). If you shine a light at the ball, the mini figure would receive less light from low angles and no light when the source is blocked by the ball. Using this logic, An astronaut conceivably could stay on the moon for twice as long as an astronaut in deep space and receive the same dose of radiation. You can read more technical details from a 2005 report by NASA Radiation Protection for Lunar Mission Scenarios.
Even with the partial protection by the lunar surface, the lack of an atmosphere and missing global magnetic fields, radiation still posses a challenge. Mapping of magnetic fields have discovered some localized magnetic fields, but much lower than the necessary amount to effectively block GCR and SPE. NASA lunar base concepts include placing moon bases under the lunar regolith. Put enough lunar rocks on top of the Moon base and the amount of GCR and SPE that gets in goes way down. Since driving in a Toyota on the moon would be difficult with a lot of lunar rocks thrown on top is not practical. As a result, Astronauts would be extremely vulnerable to the effects of space radiation during lunar excursions or cursing in their Toyota on the surface.
So the total amount of radiation that astronauts receive will greatly depend upon solar activity during the mission, their location with respect to localized magnetic fields, and the amount and type of radiation shielding used in lunar rover. The more radiation shielding, the more Mass that needs to be transported to the moon. Some concepts have proposed using water ice as radiation shielding to help protect the occupants. But once again, the more radiation shielding the more mass required to transport to the moon and the less other scientific equipment or supplies.
