Depending one what you mean by "real" motorcycle...
My first was a 1973 Honda SL70. (Some of my neighborhood friends gave me crap that the "SL" stood for "SLOW"... but did they have their own motorcycle? Noooooooooo!)
This was in 1973. I had wanted a mini-bike BADLY for a few years, but living in the suburbs, no good place to ride. Then, in the summer of '73, we moved out to "the country" (mom re-married). Part of the strategy of winning me over to the idea was getting a bike. "...and guess what Keith, you can get a MINI-BIKE out there!"
But my step-dad said No, you really don't want a mini-bike with those small wheels. He didn't ride, but was asking guys at work for suggestions. They all told him the smaller wheels would really limit the handling off-road. It didn't take much to convince me of that wisdom, and one June Saturday morning we went into Moon Motors in Monticello MN to buy my FIRST MOTORCYCLE! Brand-spankin'-new! We wheeled it out into the parking lot and the sales guy went over the controls with me. After I clumsily putted around the parking lot a little, we loaded it into the trailer and took it home.
So that's how I spent my time that summer, on that little blue Honda. Part of making the whole motorcycle thing okay with my mom was telling her... "Don't worry, I'm just going to ride it around the yard." Of course, after riding around the yard a couple of weeks, and wearing trails into the grass everywhere, I had to have more space, and started riding along the ditches and along the mississippi river banks. There were a couple of small gravel pits along my usual route, and that was a blast banging around the pits and climbing hills and using the small piles of dirt as jumps.
Funny, as nostalgic as it makes me feel thinking about that first bike, I really have no desire to get one now. If I wanted to get a bike for nostalgia's sake, I think it might be one of those dorky-looking Trail 90's. My uncle had one and let me ride it on the gravel roads by my grandpa's (this was prior to the SL70). That was like the funnest thing I'd done in my life, and I knew I had to have one!
BTW, the pic attached is not my bike, a pic from Marbles Motors of a 1971, mine was 1973. I think the only difference is the decals.