No experience or training is needed. The Zero-G folks have two very clever tricks for making the transition in and out of zero gravity much more benign than you would think. First, the aircraft has no windows, so you can't see how radically you are pitching up and down. And second, during the transitions in and out of weightlessness, they have you lie on your back, so the blood does not rush in and out of your head. It just feels like you're being pressed down onto the floor for a while, and then over a period of a few seconds, the pressure just goes away and you just gently float up. If you've ever ridden a roller coaster, this is much more benign than that.
They also ease you into it by doing a few 1/3 G (Mars gravity) and 1/6 G (lunar gravity) parabolas before going to zero G.
For me 90-95% of going to space would be to look at earth and the universe, so I would never be in the market for a suborbital joy ride, or this. I recognize that for some this weightlessness might be a large part of the experience.
I wonder if there would be much of a market for the zero-G simulation plane experience, where they have faux-windows that are just ultra high-definition video of the Earth/space. I'd have to imagine with the already strange experience of simulated 0G and the quality of video technology these days, it would be hard to tell the difference with actual space flight. Though it would certainly lose the romantic notion of actually being in space.
They also ease you into it by doing a few 1/3 G (Mars gravity) and 1/6 G (lunar gravity) parabolas before going to zero G.