Short answer:
With today's technology, assuming you're trying to minimise how astronomical the costs are, no quicker than about 5-6 months.
Long Answer:
The Earth travels around the Sun at about 30km/s. And Mars, orbiting about 1.5x as far from the Sun, travels at about 24km/s. Because of these different orbits, the relative positions of Earth and Mars are constantly changing -- they can be as near as 3 light-minutes or as far as 21 light-minutes apart.
Depending on what your objectives are, and assuming you want to return to Earth, there are only a few planetary alignments and associated trajectories that are feasible (see image below for a couple of examples). And each of these Earth-Mars launch opportunities/windows only occurs once every 2 years or so (there may be some considerable waiting time on Earth before you could even begin your transit to Mars).
The "Conjunction-Class" trajectory shows the most efficient route. Using the 30km/s you've already got from the Earth, plus whatever speed you leave the Earth at, lets say +5km/s, you could align your orbit to "grow" out to Mars. As your orbit grows, you lose kinetic energy and hence velocity. By the time you've reached Mars, assuming you've timed things right, your speed will have dropped from 35km/s to about 23km/s. This is close enough to the 24km/s velocity of Mars for it to catch you and pull you into an orbit around it. Going with these numbers, it would take about 6 months to get from Earth to Mars.
With this in mind, one could increase the departure velocity from Earth to get to Mars quicker by burning more propellant, however, even more propellant would need to be burned to slow the spacecraft velocity down to around 24km/s on arrival, to allow it to settle into an orbit around Mars.
Note, the quickest of the missions doesn't mean the fastest transit time.
So apollo 10, according to Guinness book of world records, travelled at 40,000 km/h - fastest speed attained by a manned vechicle.
Mars is 225 million km, which leads to a travel time of 5625 hours ( or 234.375 days).
If we round to 235 days, that'll take into account any potential delays (such as take off time and time taken to leave earths atmosphere).