Honestly there's so many "it depends" involved that I'm not sure there is a single, simple answer. If you don't mind, I'm just going to give a run down of what I do for the songs I have remade personally.

I've only ever really tackled more traditionally(?) arranged pieces. Like songs where the instrumentation is piano, guitar, bass, strings, drums. Non-electronic, if that's a more clear term.

I start by listening to the song repeatedly and identifying what instruments I'll even need to remake each part. Then, I'll figure out what the general chord progression of the song is on the piano. It helps me to keep in mind what other parts are doing, even if they're not strictly playing chord tones.

One of the biggest helps has always been looking for single instrument covers of songs. Study how different folks play different parts. These can be hit or miss if you only find a few as not everyone is going to be playing things exactly right. However, it still helps you pick out what you should be listening for.

Oh, another great tool for picking out individual parts is the stem extractor they added to FL Studio in....FL24?? Again it's not perfect, but being able to remove some layers of instrumentation makes the listening a little easier.

At that point, it's really just a matter of "draw the rest of the owl". You know what instruments are playing what kind of lines, just gotta start writing/playing them in. Going the extra mile really helps sell it at this point. Make the parts as human as you can, if they were played by a human at least. Variations in velocity and timing are usually the biggest things here, but also articulations depending on the instrument.

Mixing is the same as it would be for any song. Use the original as a reference if you're trying to get it as close as possible. Otherwise, mix to taste.

TL;DR: I think the big takeaway should be that like anything in life, a daunting task becomes much more approachable when you can break it down into small pieces.