Double-click the Audacity desktop icon. It may take a few seconds for Audacity to fully launch.
Click "File" and select "Import Audio." Browse for the first clip that you want to use for your mashup. Audacity supports .Wav and .MP3 file formats. The former, while larger and slower to import, has a higher sound quality. If you plan on releasing your mashup or having it played on the radio, use .wav if you have it available. When you click on the file, it will open in its own channel strip.
Click "File" again and browse for a second clip. When you select the clip, a second channel strip will open. Both audio clips will display as wave forms, with peaks and dips that represent volume.
Play both tracks together. It's very unlikely that both will have exactly the same tempo. Identify the slower track by clicking the mute button on each channel alternately so you can listen to each track in isolation.
Click on the faster track. Click "Analyze," "Time" and "Beat Finder." This command gives you a tempo read out based on the energy patterns in the sound wave.
Click on the slower track to highlight it. Click "Effects" and select "Tempo Change." In the dialog box that opens, enter the tempo of the faster track so that the tempos of the two tracks match. There should be no obvious change in pitch, or "chipmunk effect," unless the two songs were drastically different in tempo.
Listen to both tracks and select the clips you want to use. Click "Tools" and select "Trim." Click the trim tool icon onto the audio wave form to separate the sections you want from the sections you don't. Delete the unwanted sections. Once you've removed all unwanted audio, drag the audio from one channel into the spaces left in the other. If clips don't fit into the spaces, use the trim tool to make them smaller. Use the peaks of the wave form as a guide for matching beats. There should be an equal amount of space between peaks.
Use the "Fade In" effect in the "Effects" menu to smooth out the transition between clips from the two different songs.