I was part of the lucky few to be on the ZeroG plane (see the link for details).
They came to San Jose for 3 days and 5-6 flights, and I was on the 3rd flight on Saturday, which happens to also be the first time they've done 3 flights in a single day (there is a fair amount of reset time between flight, as they are required to do intensive engine checks and maintenance between each flight).

We got a little excitement before the flight, as someone forgot a driver's license and when we got there, just on time, we had to go back, as an ID was unfortunately a TSA requirement. I somehow managed to drive back from SJC to mid-sunnyvale and back in 22mn, and be back in time for the flight :)

The staff was pretty incredible, they are indeed fully committed to making everyone's experience as enjoyable as possible, and as they jokingly advertise, the flight had the most legroom available of any flight I've been on :)
In a nutshell, they take you in a modified cargo airplane, where they added padding so that you don't hurt yourself too much when you bounce off the walls ;) and after taking off and going to an area of airspace where the pilots can play, they oscillate between 24000 and 34000 feet with a 45 degree nose up pitch (which technically qualifies as acrobatic flight per FAA rules), and stop the plane, which makes it go up and down a parabola during which the plane falls and accelerates down at the speed of earth gravity, making you weightless.

This is not a new method, it's been used for astronaut training for many years, but it was only approved very recently for use by the public. It's kind of funny how it is categorized as a passenger airplane per FAA regulations, and therefore you need an ID to get in, as you get searched with a wand by a TSA official.

The flight itself was obviously very fun. After getting to the right area and altitude (and that part is a big loud because the plane, being a cargo plane, is missing its sound insulation, so I couldn't even hear the announcements from the captain), they started to do parabolas, first leaving you a little perceived gravity left (simulating marsian and lunar gravity when you can do double backflips easily :), and then 0 gravity where you can just launch yourself sideways, bounce left and right, and up and down off the walls, and drink floating water for about 25 seconds or so.
Needless to say that it was super cool, and it was over way too quickly for me, although it was about right for other passengers who started getting some signs of motion sickness (that's one of the reasons why they limit the number of parabolas)
Long story short, if I could go back tomorrow, I would :) Now I can't wait for the videos that they took of us, to see the results. In the meantime, credits go to David T for a few pictures in addition to mine, as well as the videos, which you can all find here