To remove the cables, you normally move the adjuster to slacken the cable, and it can be rotated at the top (control) end to pull free. The exception is throttle cables, which require disassembly of the right grip switches to access.
I bought a $10 cable lube tool that works really well. You put the cable into the took, then tighten it down:
You then take your cable lube (I use dry graphite lube, which sprays in wet and dries quickly, leaving behind graphite which is an excellent lubricant) and spray it into the tool:
When it drips out the far end, you're done. Careful if you use graphite lube, it's black and stains anything it gets on, just like black spray paint.