First, lets look at the distance between Earth and Mars. The closest approach of the two planets occurred in 2003, when they were only 56 million km apart. On average the distance between the two planets is 140 million miles (225 million km).
Now lets make some assumptions! Assume you either waited long enough for the next perihelion or have traveled back in time such that you can leave on time to arrive in 2003 for the distance travelled to Mars to be the shortest. Also assume that there is a straight highway to Mars with breathable air and gravitation.
So if you were to power walk that distance at 6 km/h to your favourite Beyonce tracks - believe is or not, it would take you 1065 years and 163 days! This means that you would need to leave Earth in 952 to arrive on time for the perihelion which is only 20 years after the Kingdom of England came into place.
Being a great cycling fan I would obviously hop on my bike and pedal my ass to Mars! It would still take me 319 years to get there. Better get that water bottle with me in case I get thirsty!
Seeing that we have assumed that there is a highway, the next logical step would be to get in a car and road trip there! Your trip would be 57 years long assuming that you have to obey UK highway code. Although, can you imagine 57 years without a bathroom break?
Dropping all of my assumptions, I'm sure that my predecessors mentioned everything about human space travel, in much better detail. To save our time I will skip the whole spaceship travel.
To finish off I want to give an opportunity to be a massless photon, and traveling this distance would take you 3 minutes and 6 seconds. I hope this was fun and informative, don't hesitate to ask me any questions.
With today's technology, assuming you're trying to minimise how astronomical the costs are, no quicker than about 5-6 months.
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).