The short answer is, yes, a nuclear pulse rocket could do all of that. The scales of lift capacity for such a rocket is well off the charts of any chemical rocket we have even conceived.
A nuclear pulse rocket is stunningly simple in concept. Sit a massive structure (the rocket) on a nuclear bomb and when the bomb ignites the rocket rides the blast wave into space. In theory such a system could launch the equivalent of a small city into space, and to Mars.
The mass that could be sent to Mars would be more than sufficient to allow a return trip by either a similar mechanism or by more conventional fuels. A single such ship could transport dozens of people and everything they would need to live on Mars for the rest of their lives, if permanent colonization was the goal. The time to Mars would be reduced to just a couple or few weeks.
The concept was studied in the 1960s with the name Project Orion. There are technical barriers to be sure, but the biggest problem with the concept may be legal. The proposal would arguably violate various international treaties banning use of nuclear weapons on the surface of Earth and in space.