I cut a section out a movie with Sony Vegas Pro, to get a piece of music from the film that I can't find anywhere on Youtube. Can I use Audacity to cut out the speech and effects from that section, to get only the music? If not, what software can I use and how do I do it?
John is absolutely right about the copyright implications of doing what you ask, but technically it can be done to some degree. It will take many hours to do this and at the end of the day, it may not sound the way you want it to, but audacity is capable of doing it in a very limited way. I personally do not subscribe to violating copyright so am very reluctant to tell you how to do this using audiacity, or even Magix using spectral cleaning. Better to pick a song that is royalty free and use it. There are many services out there that produce royalty free versions of popular songs that charge a relatively small fee for personal use and a somewhat higher fee for commercial use. If I knew what the song was, I may be able to point you in the right direction.
Its the medley of three of Elton John's songs from the beginning of Gnomeo and Juliet. Unfortunately, because it is a medley, and I imagine created specifically for the movie, it is difficult to get hold of.
Ok so what you are asking about is basically how to create a karaoke track. You could try audacity using this tutorial. It involves inverting one of the stereo tracks to cancel out the vocals, then using other filters to bring back the low end. This approach works well with center panned vocals. Which was common in the 60s and 70s but not so much now.
The other approach is to purchase software that is capable of spectral cleaning. Magix Audio Cleaning labe used to have it (not sure if it still does), Adobe Audition has it, as does an older adobe product called sound booth. This is a hugely time consuming process and I would reccomend that you download a free trial of any of these types of products to see if it is something you are willing to undertake as this approach will take hours.
A final suggestion is to go to http://www.karaoke-version.com/ and see if the songs you need for your medley are available there using either the Custom Backing Tracks or one of the other tracks without vocals. Their musicians are pretty good and have used their material on a number of occasions where I needed to record a piece of music where for example I just needed a drum part, or dums, keys, and synth...etc.