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There are a lot of different small engines that work really well on bicycles and a lot of it depends on your budget and your skil set to which ones you'll be able to use. If you got a good sized budget for your project, you could use a DiNardis or Morini engine which is about 6 to 10hp for a 50cc engine and close to 15hp for a 65cc, also depending on whether the engines are air cooled or liquid cooled, the liquid cooled engines can put out more power, but the air cooled ones would be a lot easier to get installed.
There's also a lot that can be done with a China Doll engine and these are the easiest to install into a bicycle frame as well as the least expensive. These engines are around 2hp stock and can be modified to make around 5 or 6 hp before they start to get a little expensive. There are a few that make racing engines out of these and these can put out around 10hp set up as a racer, but by the time you got one to that level it would have costed less to buy and install a Morini or one of it's clones so there's a bit of a judgement call to make when shooting for above about 6 hp since you can buy a 10hp China Doll race engine for around $1100 or you can buy a 9hp KTM clone for around $500 etc.
The 4 stroke world is about the same, you can buy a 212cc Predator engine for just over $100 and it's going to make 6.5hp out of the box, but you can spend up to and above $1000 on these engines and get close to 25hp out of one with the right parts, machine work, and tuning. Then there's a decent selection of 50 to 150cc motorcycle or minibike engines that'll fit in a frame with some modifications, and these engines can be had for a few hundred bucks or so.
The transmission really depends on the engine selected and whether you want a few gears or just want to run a single speed. Single speeds are lighter and simpler but with single speed, you have to set up the gear ratio either for nice acceleration or good top speed, or any compromise in between. single speeds can work really well tho. One of the simplest ways to have better acceleration and a decent top speed is to add a CVT type transmission or you could even make a 2 or 3 speed by using centrifugal clutches and jack shafts or run a jack shaft and use the bike's deraileur gearing.
The possibilities are definitely there, it all depends on what type of budget you have, whether or not you can weld, and what you're looking for to do with your project.
