I don’t think there are any plans to send humans any time soon back to the Moon, so really your options are limited at this stage!
But you might not need to go to the moon to experience zero gravity — you could be a space tourist, being launched into low-earth orbit on a reusable rocket. This would give you the free floating experience that you get in space.
Such commercial space travels will be soon available from companies like Virgin Galactic. Right now, you have to have lots of money to afford it (something like 200,000 quid!) but soon prices will drop and normal people will be able to afford it. So I think you should be able to fullfil this dream during your lifetime!