I was fortunate to catch the Star Trek Science exhibit in 1994 at the Denver Museum of Natural History. There were many great interactive exhibits on sensors, tricorders, the ship computer, warp drive and more! Unfortunately, I do not think the exhibit is travelling any more.
However, it looks like we've finally achieved another product of Star Trek Science: Transparent Aluminum! Prominently featured in Star Trek: The Voyage Home, transparent aluminum was widely used to make the windows in the Starships in Kirk's era, and to make the holding tanks to transport humpback whales to the 24th century!
The real science is pretty interesting. Check out the article here: http://physicsweb.org/article/news/8/8/9.
Here are some other interesting links about Star Trek Science:
See also: Sci-Fi