Tools
If you plan to do this again in the future, get good tools. It'll be frustrating if you try it with hardware store diagonal cutters, they mash the internal wire. And cutting the housing would be difficult.

Park tool cable cutter. This makes very clean cuts in the wire, and easily cuts the housing, but then you'll want to grind or file the housing end flat and smooth. The spiral brake housing will have the end of the spiral mangled, so it needs to be smoothed. The shifter housing needs to have an even, 90 degree cut of all it's internal reinforcing wires, and this often needs trimming after cutting. And the cable cutter crimps the cable ends effectively, too.

Dremel grinder to easily smooth the cut end of the housing. And it has lots of other non-bike uses, too. You could use a file, but the Dremel is fast and easy.

Some BF posts claimed to use the Dremel cutoff wheel to cut the internal cable wire. I tried this as an experiment, and it just frayed the strands completely. I don't see how they could make it work.

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You'll need to know how to adjust your derailleur shifting after the new cable is installed.
I start with the adjusters screwed in most of the way, clamp the cable into the derailleur, then unscrew them part way. Then adjust from there.

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My local bike shop sold me bulk Jagwire housing off their 50 foot spool, and two internal shifter cables.

I like doing it myself, so I can cut the under the tape housing to the "just right" lengths. I cut it approximately correct, then insert one end into the shifter, and use temporary electrical tape to hold it against the bars, then mark the other end at the frame stop. A second cut makes it fit nicely.