What size/engine/payload aircraft would be needed for carrying two passengers and a motorcycle? I'm not even in the realm of knowledge of aircraft so don't chew me out on lack of terminology please.

My objective is to fly myself and a 300-pound motorcycle to destinations to ride from there. 

I weigh 185 lbs, the motorcycle weighs 300-400 lbs, any secondary seating and other interior furnishings would most likely be removed to reduce weight, and, obviously, all the necessary components would stay. As a side note, the aircraft needs sufficient enough door to get a small motorcycle in. Let's not even talk about how unsafe this might be; this is all hypothetical. 

How large, or what class, of an aircraft would be needed for such a task to be completed?
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)