Open the audio-editing program. If you do not have one, download a free audio editor such as Audacity.
Open the song that you'd like to become the first in the final combined file.
Click "project" on the uppermost toolbar, then select "import audio" from the drop-down menu. Choose the second song for your mix. The song will open as a simultaneous track to your first choice. Click this second track to highlight it.
Select the "time shift" tool (line with arrows pointing left and right) from the toolbar. Click and "hold" your highlighted track and drag it to the right until the beginning of this second track lines up with the end of track one.
Click the "cursor" on the toolbar. Move the cursor to a place shortly before the end of track one. Click "play" on the buttons toolbar. You should hear the second song begin as the first one ends. Click the "stop" button.
Adjust the second song to the left or right again if necessary. Once you're satisfied with the mix, click the "rewind" button to move the cursor back to the beginning. Select "save file as" from the file menu and pick a memorable file name and location. Play the new file to make sure it saved properly.