Decide what you want to mash together. Brainstorm individual songs, albums or even just melodic lines or beats that really inspire you, and then gather data on them including the chord progressions and rhythm to compare to other songs.
Set a tempo grid using a digital program like Tracktion, or any digital audio program that you can use to make music and combine tracks. Once you've set the tempo, you can fit your different tracks to the same tempo to combine them.
Copy the individual tracks (like bass or vocals) that you want to use to make them new tracks. If you don't have a program that will do this for you, some are available for download from the artists themselves, so look into artists like Nine Inch Nails, Radiohead and others that have sold their tracks in the past.
Combine different elements from different songs to create something new. Try to create a new track without adding any original work to make a true mashup. What differentiates a mashup from a remix is that you don't just change the tempo or the rhythmic line, you actually add those elements from external sources.
Tweak rhythms and pitches after you put the product together. It's hard to actually create a mashup when you're worrying about fine details as you go, so wait until you're finished to make sure that beats and pitches match the rhythm and key without any noticeable outliers.