Restoring/modernizing an old cheap(ish) bike tl;dr;? Just see bullet points below...

My mom recently gave my wife her old Raleigh Horizon bike.  She must have gotten it around the year 1990, and there's no way she spent much on it (she's kinda cheap).  

I don't know much about bike's - I mean I rode them every day of my life as a kid, but I never really got into the nitty gritty details of a repair or upgrade.  My plan with this Raleigh, is to repaint it, which shouldn't be a problem.  However, I also wanted to replace the cables and shifters if possible since it's disassembled already.  Possibly the cassettes, too.

Finding information on an old cheapo bike like this is kinda hard.  Most Youtube videos are for hardcore guys who are all about their $1000+ bikes (probably you guys lol).  

I basically want to get my wife a bike that works well but isn't going to cost me more than $100 (USD).  I have the bike, it technically works, but the chain does slip off when shifting a lot and these shifters need to pressed pretty hard to make them do anything.  They're also slightly rusted.

I've already disassembled the entire thing to prepare it to be painted.  I just want to reassemble it with some new parts to make it better...

My questions are:


Are brake cables and shift cables the same? On Amazon I see most are sold as "shift" or "brake" cables but they seem the same to me...
What type of cables would an older bike like this need if I wanted to replace them (and maybe upgrade, if not too expensive)?  
Can you recommend a set of sprockets (or cassettes) that would allow the shifting to work better.  Nothing extreme, just simple but decent ones... if that's a thing :)
Can you recommend a set of shifters?
Is there any other inexpensive upgrade that you might recommend?


Thank you!  Sorry for the noob questions 

UPDATE: updated the terminology to be correct (I think).  Thanks @ojs :)
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.