How would it be done?
what kind of craft design would be needed? what kind of drive is most plausible?
Exactly how you do it depends on what you're taking, and how much of it. What kind of ships you need to move 100 people and a million cells may be different than what you use to move 10,000 people.
The absolutely most plausible drive would be what we use today - chemical rockets. Lots of them. Big ones.
The next most plausible is a plasma drive, which NASA already has in development. These use electricity to ionize a gas (hydrogen or helium, in most versions), and accelerate it out as a rocket.
Everything else I'd call theoretical.
How long would it take with present technology?
The travel time depends on when you go - both planets are moving, so when you leave matters. For chemical rockets, there's an absolute cheapest way to get to Mars, for which there's a launch window every 25 months. You put your rocket into a "transfer orbit" to Mars, and it takes you about 214 days to get there.
If you have effective plasma rockets, which may be able to put you under constant boost (which chemical rockets can't reasonably do), then the math gets a little complicated, and how long it takes depends on the details. Less than 219 days. One plan for VASIMIR plasma rockets would get you there in about 40 days.
"Gold foil and gamma shielding"? No. You use water (ice, frozen in tanks around your crew capsules, very probably) as your shielding. You'll need to take it along with you anyway, and it absorbs radiation pretty darned well, so you might as well make it work double-duty.
Your ship is probably a long stick, with an engine on the end, and pods of cargo and living quarters bolted on along the stick.
Humans drop down to the surface like they did for the Moon - on top of a retro-rocket. Mars' atmosphere is too thin to use wings for effective breaking and a drop from orbit, so no "shuttle like craft". Non-fragile cargo might get rougher treatment.
Also, what would become of the moon?
As I mentioned upthread, literally destroying the planet is not going to happen from a standard asteroid impact. The surface of the Earth may be uninhabitable for the foreseeable future, but the planet is still there.
So, the Moon is still there, orbiting a dead world.
Now, if you're lucky, and the Earth isn't really too screwed up, you might choose to lift to the Moon instead of Mars, in the hopes of coming back in a little while. It is a *lot* cheaper to go to the Moon - you don't need nearly so much fuel, you need less shielding on the trip, because you're still covered by Earth's magnetic field, and so on.
But, there are also drawbacks to the Moon. For example, the lighter gravity may pose major developmental problems for children (Mars might have the same issue, but to a lesser degree). And best guesses are that Mars is more rich in metals than the Moon, so getting resources may be easier. Also, even though it is thin, Mars has at least some atmosphere, which gives a few advantages as well.