While there are many programs available that are specifically designed to automatically split an mp3 into multiple tracks, consider learning how to edit sound files themselves. This will be a more versatile and ultimately more valuable approach than using a specialized program.
You will need to obtain a program capable of digital audio editing. There are many commercial programs used in professional recording that will suit your purpose, though there are also some more basic programs (GarageBand is one such program, which comes standard with many Apple computers; Audacity is a cross-platform open-source audio editing tool). In a sound editing program, you can open a file and visualize a song's waveform. This will allow you to manipulate an individual file.
Open up the mp3 file in your software. You will see the song's waveform, which allows you to visualize the dynamic range of the song. Treat the song as a text, from which you will be cutting portions. Locate the first desired track, and select it with the program's selection tool. You will then be able to edit the track as if you were editing a document. "Cut" out the first track.
Now, you will want to create a new audio file within the program (don't save your changes to the original, in case you make mistakes). Once you are in a new file, select the "Paste" function, and your selection should appear. If desired, you can then edit the new track. For example, it is often useful to apply fade-ins and fade-outs to the beginning and end of a track, respectively.
Once you have finished editing the first track, save it as a distinct file and then repeat this process by opening up the original mp3 for as many tracks as you like. Notice that this is also a valuable technique to use if you need to cut up a song that is too big to send as an attachment or too large to fit onto an audio CD.