Why an Apollo-style mission at all? What about starting with an ISS-style space station in orbit around Mars. Send the pieces over a few years. Have them self-assemble (if the technology allows) or send a construction crew out to put it together. You can do a lot of exploration with near-real time control of rovers and the like. If you really want a human presence, spend time to really scout out an optimal location. You can send landers separately from people, load em up at the space station and send them down. Use rovers/telepresence to build a habitat before sending anyone down, sufficiently stocked so that the slightest glitch isn’t a death sentence.
Only if you don’t care about the health of the astronauts. First you have a long flight there (nine months for a Hohman transfer orbit), exposed to cosmic and solar radiation and in zero g (these are bad for you for extended periods of time) and then they stay in a station with yet more exposure. Followed by a nine month trip home with, yep, even more. These guys are going to be basket cases by the time they get back to Earth.
You could, I suppose, eliminate the zero g with a revolving station (although for some reason no one’s ever even tried to make one), and add shielding to reduce solar radiation, but cosmic radiation isn’t so easily reduced. It’s much more energetic, so goes right through shielding that mostly blocks the Sun’s particle radiation.
The best shielding for cosmic rays is none at all. It goes right through nearly everything, and that includes humans. Radiation going right through you isn’t a problem. What can be a problem is when a cosmic ray hits an atom of your shielding, and sets off a cascade of many other lower-energy (but still ionizing) particles, which don’t go right through a person.
And nobody’s ever built a spinning space station because, so far, the purpose of space stations has been to study the effects of zero-g on humans and other specimens. But we really ought to build a spinning one, because there’s almost no data on the effect of low (but nonzero) gravity, beyond the few days that astronauts have spent on the Moon. And that’s something that we’d really like to know, before we send humans on a long-term mission to the Moon or Mars.
But you’re still going to get that effect (at least a little bit) from the walls and equipment in the station. It may be reduced without additional shielding, but then you have no protection from solar storms.
I think the first thing needed is a base on the moon; land prefab pieces (sort of like the ISS) and assemble them, cover them with a layer of dirt to shield from the worst radiation, the dirt is there for the taking. The first bit can be done remote-control. Using what we learn there, we can do the same with the Martian moons, we hope. Instead of a big bunch of exposed tin cans, maybe burrow into the moons for protection. Of course, we’d have to find some sort of tunnel-boring technology that could be sent to Mars. Same with the landers; the Mars landers would be either human shuttles - small, reusable to move people up and down from the surface - or cargo landers, more likely able to land a heavy load but then leave it there to become part of the installation. Cargo landers would also take off again, but carrying a lot less load - maybe rocket fuel and supplies for the moon stations. If the regular ships from Earth don’t have to also carry along lander engines and fuel for landing their cargo, so much the better.
You don’t take off in the quite same vehicle you landed in; you take off in the top half of that vehicle, in a module known as the Mars Ascent Vehicle (MAV). The lower half of the landing vehicle acts as a launch pad. But I’m sure you know that. And of course you are right; both landing and taking-off on Mars are far more challenging than landing on the Moon, because of the gravity.
One interesting possibility that NASA has explored is the collection of water on Mars for propellant (In Situ Resource Utilisation).
This would lighten the mass of the landing vehicle/MAV considerably, and make landing easier. Since Mars has a very thin atmosphere, aerobraking is much less useful than on Earth, so you have to use a lot more fuel on the way down to ensure a soft landing. If you can collect propellant on Mars, that would make the mass that needs to land softly significantly smaller.
ISRU is a risky strategy, however; if the astronauts get to Mars and can’t find enough water to launch, they are stuck there and will die. This is not without precedent in the annals of exploration, of course- the Franklin expedition of 1845 took enough resources for four years in the Arctic; by 1850 they were all dead, having failed to perform any significant ISRU.
For a one-off Apollo type mission, perhaps the first one or two, the two-piece lander that leaves behind the landing tanks is probably Ok. For the long run, the ideal would be to have one-piece landers; refueling originally in orbit from fuel shuttled from earth (meaning the next few missions don’t need to include a lander) and hopefully eventually refueling on Mars to shuttle fuel up to orbit for the Earth return.
Our space missions until ISS basically had the same problem - the entire mission had to be launched at the same time, and then the equipment was discarded afterward. The key to progress is to produce reliable reusable equipment, send it where it needs to be once, and leave it there to be re-used.
If you’re going to use ISRU (which is pretty much a must, for Mars), you’re not going to do it with astronauts. You’re going to first launch machinery to make fuel (or whatever else you’re making, possibly even including food) and store it in tanks, and then only once you’ve confirmed that the machinery worked and the tanks are full, only then do humans leave the green hills of Earth.
How much extra protection from cosmic rays does the atmosphere of Mars provide? Of course you could tunnel into the surface but you’ll need boring equipment for that - and, as usual, Elon Musk is ahead of us all.