The magnetic field protects only against charged particles (usually coming from the Sun). Galactic cosmic rays or solar flares are not blocked. The atmosphere blocks most of that, as well as the planet itself. In low-Earth orbit, you have half the horizon covered by the planet, so you get only half the radiation.
On Mars, you would get a little lower radiation as you would in low-Earth orbit. The atmosphere albeit thin does provide some shielding and the planet itself removes half the radiation (as in LEO). Greater distance from the Sun also reduces the impact of solar radiation.
Normal radiation levels pose little danger in space, you mainly have to worry about shielding during solar coronal mass ejections. When that happens, you usually have a few minutes to hours of warning (courtesy of our fleet of solar observing satellites). For adequate shielding you need to design a 'storm shelter' in the middle of the spacecraft, surrounded by the crew's food, water and waste.
Note that we've had astronauts spend years in space and the physiological damage was from low gravity rather than radiation. I don't understand why NASA has never built a rotating space habitat to solve this problem.