Restoring/modernizing an old cheap(ish) bike
There's a lot of questions in here. My suggestion is to read through https://www.parktool.com/blog/repair-help and http://sheldonbrown.com/ instead of asking a separate question for everything.


Brake cables are thicker than shift cables, and the outer cables are constructed differently. There are different variants of the cable ends for both, you need the correct one. "An older bike like this" is not enough information to guess which one you need, there are a lot of different old bikes sold as "Raleigh Horizon".
We don't really do product recomendations because they tend to be location-specific and information gets out of date. Once again, we need more information to tell which cassette or freewheel fits your bike.
The above applies for shifters, which I guess you mean by "shift nobs".
If the tires and brake pads are original, replacing them with new decent quality ones is a huge improvement.


What you didn't ask but I'd recommend anyway: lubricate the chain and cables and adjust the gears before buying anything. The shifters are probably stuck because of old, dirty grease, and cleaning them with WD-40, CRC 5-56 or similar should help.

Update after pictures: This style of shift lever can be completely disassembled and greased. If you want to replace the sprockets, what you need is a 6-speed freewheel and appropriate tool for removing the old one. New one can be screwed on by hand. Newer freewheels have teeth shaped for easier shifting, so upgrading may improve shifting. If you change to a new freewheel, you need to replace the old chain too, because it has worn together with the old freewheel. The correct cables are MTB-style for brakes and road-style for shifting.