Sounds like you're in the market for a brand new torch ship! Step right this way...
http://www.projectrho.com/public_html/rocket/torchships.php
<PERSON>'s site goes over various definitions for a torch ship, but it boils down to any rocket which has both high thrust and high specific impulse (fuel economy). In general this means getting away from chemical rockets and moving over to rockets which use some kind of nuclear reaction.
Ones we might be able to build in the near future are mainly based on fission - Orion drives and nuclear salt water rockets. These are every bit the radiological hazard you might expect them to be, so you wouldn't want to operate them in a planet's atmosphere. Not one that you cared about, anyway.
More speculative versions are based on fusion. Some of these (deuterium-tritium rockets, for instance) are still pretty dirty. Others (proton-boron and proton-proton) are much cleaner, but you still wouldn't want to be anywhere near the exhaust when the engine was running.
Naturally, the cleaner reactions are much harder to initiate and sustain than the dirtier, more neutronic ones. Thanks, universe.
All of these have the potential to get you to a closer planet like Mars in pretty short order. The exact amount of time will depend on how much delta-V your ship's engine and fuel fraction gives you, but think in terms of days instead of months.
This has a lot of advantages. 199
For one thing, shorter trips mean less radiation exposure for passengers and crew. Engines that aren't as mass-limited mean that you can push more cargo and carry more shielding. The spacecraft is reusable, which keeps costs-per-trip down. You can reach the asteroid belt in only a little more time than it would take to reach Mars, so you can potentially extract raw materials from there to maintain your space effort, instead of having to lift them out of a gravity well.
It's not all roses though.
If you're using an engine with a highly radioactive exhaust, you'll probably only want to use it in space. That means that you'll need a more conventional way to escape Earth's gravity well to get to your interplanetary craft.
Fortunately, that falls into the realm of reusable rockets (now not actually science fiction any more) and spaceplanes. Check out the Wikipedia page for SpaceX for a breakdown of their current and future reusable launch vehicles, or Skylon for an idea of how a spaceplane might operate.
Project Rho is a veritable gold mine of engine ideas, as well as in-depth discussions on any other facet of spacecraft design you can think of. All of it's are based in real physics, too. Scroll down to the bottom of the page for a full list of subject areas.
Everything listed represents a significant engineering challenge, but none of it is physically impossible. 199