You could use a dirtbike or pitbike or quad engine, but if this is your first project then I would use an industrial engine like a Briggs and Stratton or Honda, like the ones that you saw on the go karts you went on. These are not usually found on lawnmowers as they are horizontal motors, but have similar power ratings.
Do you have a harbour freight near you?
They sell a very reasonable engine that you can get for 99 dollars if you use a coupon. It is a copy of a Honda gx200 and is suppose to be 6.5hp (it's really about 5.5hp, but it can still shift you ok) you may be able to find a cheaper used engine like a Briggs 5hp or an old Honda on craigslist or eBay. These industrial engines are probably the easiest to start with IMO. Here is a link to the hf engines. The 6.5hp is fine, bit for more money and power you could use the 11 or 13hp, any of the engines in the link should work, except the 5.5hp lawnmower one at the end.
http://www.harborfreight.com/engines-generators/gas-engines.html
You can then get a clutch, chain, sprockets axle and wheels from somewhere like bmi karts and buy or find some steel, and weld up a frame, or buy a used frame from craigslist or get one from Azusa engineering.
Rob