Download your chosen audio editing program and install it. Most of the free audio editors are similar in operation, displaying a wave-form version of the tracks that can be manipulated in a variety of ways.
Choose two songs to mash up. While choosing different genres of song can create the most interesting mixes, it is easier if there is some thread of similarity between them. Look for songs you can find an a cappella or instrumental version of, to make the mash-up an easier process. If possible, find songs that are similar in tempo or have matching keys. This can be changed later if they are different.
Load both tracks into your audio editing program. On Audacity, the first track can be added by pressing "Ctrl" and "O" on your keyboard, and the second by going to the "File" menu and selecting "Import" followed by "Audio." These tracks will both load as separate stereo tracks. Add two new tracks to make your mash up on.
Determine the key of one of the songs. This can either be done by looking for the sheet music online or working it out with the aid of a keyboard or other instrument. If the songs are both the same key, you can skip this step. Change the key of the other track to match the first song's key. For example, change the a cappella version of "Hit Me Baby One More Time" by Britney Spears to the key of E minor to fit to the instrumental version of "Enter Sandman" by Metallica. Highlight the song and select your program's "Pitch Editing" function. On Audacity, this is found under "Effect" and "Change Pitch."
Match the tempos of the songs using the "Change Tempo" function on your audio editing software. If you don't know the specific BPM (beats per minute) of the songs, you can just slow the tempo of the faster song down or speed the slower song up to match the tempos. Use trial and error to get the correct tempos. Look at the peaks on the wave form to understand the basic beat of the song.
Select vocal parts from the a cappella song that you wish to insert over certain instrumental parts of the instrumental song. Place the instrumental track on one of your blank tracks, and then layer the vocal on the other free track as required. Do this using "Cut" and "Paste" on your music editing program. Preview your mash-up and make any additional changes required.