I worked as an undergrad doing research in soil physics at USU. The professor I worked for had a project funded by NASA. The general idea of the NASA funded research is to optimize root zone sysems for growing plants efficiently in space. Most of the plants grown in space have so far, not intentionally, been miniaturized versions of the same plants here on earth. This is because the roots have been drowning or close to it in too much water and not enough air or not getting enough water.
We designed experiments to be flown in a zero gravity environment to test some theories regarding fluid movement. You can see more details about the research here
With those experiments, I got to go to Houston and fly on the Vomit Comet out of Ellington field in February (2003), June (2003) and again in Feb (2004). It is a thrill to be weightless and then comes the moment when the plane pulls up.
Short of flying on the Shuttle or the Russian Capsule the only way to really get zero gravity is this way. NASA's Reduced Gravity Program on the KC135, commonly known as the "Vomit Comet". It is a special plane that flies up to about 33,000 feet and drops, well mostly. It creates roughly 25 seconds of zero gravity and then they do it again. You can see more details about the flight pattern and the KC135 at, of special interest maybe the flight trajectory page: Nasa's Reduced Gravity Center
NASA let Ron Howard film Tom Hanks and crew floating around in zero gravity for the film "Apollo 13". There are no wires attached to them making them float, it is real.
Even better than Apollo 13 is the I-MAX Space Station movie. Wow! I saw it at the Clark Planetarium in SLC. It gave me flash backs to being weightless on the Vomit Comet. If you have ever thought about being an astronaut, the Space Station movie will make you want to send your application to NASA. You can find those applications at: Nasa's career page