Recent radiation measurements from one of the landers/rovers during a solar storm showed that even Mars' anemic atmosphere shields from radiation much better than previously thought.
Enough that it mitigates a lot of the need to dig, at least right away. A permanent or long-term base would want to do so anyway to get as much or all of the radiation as possible. But whatever landed capsules or modules + atmosphere would be adequate, or even just a little soil on top of the modules with some baskets or mesh berms to hold it would do. Some designs also add shielding mass by sticking water tanks and as much gear in the ceiling/roof of the module as possible.
As far as the trip to Mars, the better the propulsion, the less flight time, the less radiation exposure.
To do that would mean some sort of nuclear propulsion, either one of the various direct nuclear rocket schemes, VASMIR or one of the other plasma thruster schemes, or ion engines powered by a nuclear reactor. Also, if the nuclear reactor is big enough and has spare watts, an electromagnetic shield could be produced by placing current through a conductive cage around the habitation modules of the ship would deflect charged particles from a CME just like Earth's field does. Although that would take a LOT of current. But maybe someone would get clever and send a extra-large reactor for a ship on a one-way trip that then gets used at a Mars orbit station, or maybe land it for base power, and it would have the Watts to spare....
Neutral particles could be handled in a couple of ways. Although in some circumstances with radiation, "not enough" shielding is worse than none. Really energetic radiation like cosmic rays, it's better to not "fight it" because if you try to shield from it, and it's not enough, it just creates more atoms for them to knock into, and create unhealthy secondary particles/radiation, while the original cosmic ray could zip right through a "thin" ship and your body and not affect anything. (Apollo moon astronauts reported being disturbed or woken from naps from the flash of the occasional cosmic ray passing through their eyeballs vitreous humor, or maybe directly exciting the optic nerve...
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To save weight the reactor might be on a long boom or stalk, or have what's called a "shadow shield" that only shields the direction of the crew module, and all that could maybe shield them from a CME the reactor and it's shielding would be pretty dense/massive anyway, and orienting the ship so it's in the "shadow" of the most mass might work.
Or anything with lots of protons/Hydrogen would work well for neutral particle shielding. A "storm shelter" in the shadow of some water tanks, certain polymers, or even a liquid Hydrogen tank if the ship's design called for it. Although if the Neutrons are high energy enough, secondary gamma radiation can be produced, so again it's that balancing act that rears it's head again.
