What size/engine/payload aircraft would be needed for carrying two passengers and a motorcycle?
The question is incomplete without information about the range this aircraft should have. Pilatus in Switzerland is always fond to tell of one of their customers who bought not one, but three PC-12s for exactly this purpose, and has one plus a Harley-Davison motorcycle stationed on each continent of interest. The size of a PC-12 allows the installation of a crane, so loading and unloading of the motorcycle are simplified.



PC-12 side view, with open cargo door (picture source)

Obviously, even a PC-12 does not have the intercontinental range to go anywhere, but if you plan to stay within one continent, range is not an issue. Alternatives in this class are the Czech Aero 270 or the Cessna 208 Caravan. All use a PT-6 turboprop engine, and you will need professional maintenance to keep them flying, so using them does not come cheap.

A more reasonable alternative would be piston-engined, and here the Cessna 206 or 207 (which @Terry already suggested) would be a candidate. However, fitting the bike in will be a challenge.

If you want more internal space and are not afraid of antique taildragger airplanes, the Antonov An-2 or the Canadian DHC-3 Otter would also be a good and reasonably priced choice - however, maintaining one (and especially its radial engine) will also burn a hole in your pocket. 



DHC-3 in Pensacola (picture source)