The essence of a song (though not necessarily an electronic track) is the vocals - the lyrics and the melody. If you are taking a conventional song and remixing it in a dance style - probably the most common way to approach this whole exercise - you can throw away almost everything but retaining the original vocal, even if you process it differently, will mean the track remains recognisable to listeners. You can even replace all the music, but keeping the vocal is a good way to anchor the track. Mess around with it too much and your remix might be written off as a completely different track.
This may be out of your hands but if possible, ask the person supplying the original material that you plan to remix for versions of the stems (the individual, project-length audio files that make up the mix) in both dry and wet (with effects) versions. This will enable you to fully control the texture of the remix, rather than having to contend with reverb, delay and other effects that they have already applied when mixing. Of course you may decide to go with the wet versions, but it’s nice to have the choice.
Some DAWs are particularly good for remixing. Ableton Live is one example, with its looping and time stretching abilities making it especially suited to chopping up and manipulating sections of existing tracks, or stems, to reinterpret them. In this course "The Art Of Remixing" from expert producer Olav Basoski you can learn all about the details of remixing in Live, and how to get the most out of its remix-remix-targeted features like looping, slicing, editing and arranging whole new tracks from existing material. https://ask.audio/academy?nleloc=course/2530/the-art-of-remixing
There are of course many ways to approach a remix - you could make minor changes, strip away a few elements and enhance others. Sometimes though it’s really interesting to go all-out with a treatment, radically altering the original track in ways that bear little or no relation to the original. Drastically alter the tempo, the key or the arrangement. Turn an acoustic ballad into pounding techno - whatever works. The great thing about remixing is how much freedom you have to experiment.
The amazing thing about digital technology as opposed to analog tape is that you can just keep going and going. So if you have done a cool remix of someone else’s song, pass your remixed stems on to someone else to reinterpret (provided you have permission from the original writer). See what they do with it - the sky’s the limit.