I watched that! I was so excited to see it on, although they did cut a few parts out. If you're going to watch it without seeing the episodes, there are definitely a few things you should know.
(Spoilers!)
River was taken by the Alliance because of her incredible abilities. She's a genius, and she's also essentially psychic. They performed experiments on her and messed with her brain, and she's resultantly... well, crazy. Simon knew she was being tortured, and he gave up everything to get her out. Now he acts as doctor on Mal's crew so that he and River can keep moving and stay out of the Alliance's reach.
Mal and Zoe fought in the war against the Alliance, and they lost. The Alliance has since taken over, and Mal hates it; as a result, he lives on his ship Serenity (named after the battle that ended the war) and finds work wherever he can. Zoe is fiercely loyal because she fought side-by-side with Mal, but she's also married to Wash, the pilot. Jayne is the hired hand, and has betrayed the crew in the past. Kaylee is a brilliant mechanic, and she and Simon have something between them, although nothing's really happened. Shepard Book traveled with the crew throughout the show, and he has an interesting past that hasn't been explained. Oh, and Inara is a Companion, meaning she's paid to sleep with men which it an honorable profession in the future.
As for the general set-up: New planets were terraformed to replace earth. Some of them are really rich and fancy, and have the technologies you'd think would exist in the future. Some of them, though, are like something out of an old western movie; poor and struggling.
That should help a little, although I'd say it's really better to watch the show and get a feel for it first; otherwise the movie's a little jarring (I watched it first and a lot of things went over my head). Also, you're more attached to the characters and the story; once I went back and watched the Serenity again after seeing the show, it was almost a different movie for me.