helpful links...
A computer search for "production music" will show you hundreds of companies offering music. Most of those are aimed at the trade and their rates are often prohibitive for non-commercial users. Those listed here are relatively modest in pricing and "n-c friendly". Almost all of them allow you to listen to part or all of a melody in low quality without charge. Do check the license terms for any music or sound-effects you download. Even if there is no fee you may find a request that the composer or source be cited. Please honor such requests.

Remember that most such music is intended for background atmosphere rather than as a "main theme" in a movie. That said, you can often use a specific theme for each character in a story and blend those into satisfying music sequences. It is much rarer to find a piece of modestly-priced royalty-free music strong enough to play on its own for several minutes under (say) beautiful scenic views.

Note that most music does not include Performance Rights, which may apply if you show your work in public rather than within private homes or your film club. Links to the various agencies concerned with such rights are below.
To the best of our knowledge these sites are all safe, but you should always check any file you download with your anti-virus system before opening or running it.
2-b Royalty Free - huge stock of music in a very wide range of styles. Good search faclities. Fees mainly between $5 - $35 a track.
AKM Music - a British-based company whose range includes "Performance Rights free" music. Keyword search. The catalogue includes a "Cine to Video Selection CD" with longer tracks and organized music loops which can be fitted together. Between $5 and $14 a track.
Audio Network - huge library with separate sections for prodiction music, classical and sound-effects. Use their "Home/Student Productions" license. Priced in British pounds but tracks are around $1.30
Trackline - wide range of music. Buy CDs or tracks. For example 5 minutes of Bach for about $10
Incompetech - better known as the home of Kevin McLeod who makes loads of excellent music available free - optional donations welcomed. All he asks is a credit. If you cannot give a credit there are modest fees.
The Music Bakery - has a wealth of music of all sorts. Fees vary according to length of track but 4 minutes for $47 is typical.
NVM - they sell CDs rather than tracks. Around 60 minutes for about $28.
Bobsongs - Bob Chambers often produces tracks between 3 and 6 minutes long. He prices in British pounds but tracks cost about $2 or you can buy an album at about $22 or a bumper bundle of albums for around $56.
Fotosearch - despite the name this has a good music section too. Wide range of prices.
Primary Elements - they offer various licenses. For most of us type A is appropriate which costs $3 a track. CDs often available at bargain rates.
Royalty-Free TV - their standard license suits most of us. Full length main theme $29.50. Lower prices for short clips.