Reddit mentions of KMC Z50 Bicycle Chain (6-7-Speed, 1/2 x 3/32-Inch, 116L, Dark Silver/Brown)

You don't need a 21 speed chain, you need a 6/7 speed chain. Here's an example of this. Most stores which sell bikes, including department stores, stock these chains. You can pretty much use any chain tool . Some choose to use a dedicated chain tool, while others choose to use the chain tool in their multi tool. If you don't already have a bike multitool, I would highly recommend picking one up. This one is very highly regarded and the additional $11 over a generic chain tool might well be worth it, if only because of the huge amount of added functionality.

For tips on changing a chain, you can watch this video.

>Here’s the kicker: it needs tires, tubes, cables, and a chain.

Going by the photo I wouldn't be surprised if you could get away with lubricating and reusing all the existing cable housing, just replace the inner cables, which could reduce costs a fair bit depending on how much you pay for housing and make the job quite a bit easier/not needing special tools you need to cut housing

If the tubes hold air they don't need replacing unless you have problems

Have the tires actually perished? They might still be OK to ride for more casual riding, though new tires would perform better and be smart if you were doing more serious stuff

It's even possible the chain could be derusted in an acid or evaporust bath and would work OK after lubrication and freeing up, but considering how cheap chains are from Amazon etc I'd just buy a new chain like this one
https://www.amazon.com/KMC-Bicycle-6-7-Speed-32-Inch-Silver/dp/B074YZGKKQ and a basic chain tool

Definitely a good decision by attempting to do the repair yourself instead of taking it to the LBS, you'll be saving a lot of money.

You can get the tools and parts for cheap on Amazon. you'll be paying triple+ if you buy them from the LBS, though the trade off is you won't be helping support "the little guys".

i'd hold off on replacing the front chaining since they wear at a verrrry slow rate compared to rear cogs, and only do so if you have similar problems after riding on a new cassette & chain.

You'll need a chain breaker tool to get the old chain off, this is one I purchased about 2 years ago, used it on 6 or 7 chains and it's still is going strong, under $10 - https://www.amazon.com/gp/product/B0179JC31I/ref=oh_aui_search_detailpage?ie=UTF8&psc=1 and a new chain- https://www.amazon.com/KMC-Bicycle-6-7-Speed-32-Inch-Silver/dp/B074YZGKKQ

also when was the last time you had your rear hub repacked? i'm not sure if just slapping a new cassette on that wheel given its state is a good idea, you might want to consider repacking with new grease and bearings and lubricating the free-hub first, which is more involved.

personally i'd say either get a new rear wheel, chain & cassette, then tinker around with the old wheel on the side, experiment with it, and ultimately keep it as a back up, or take the wheel to the LBS and let them overhaul the hub / lubricate the freehub before putting a new cassette on yourself, unless you want to spend the time learning how to do all that yourself, on-top of having to buy additional tools, bearings, grease, etc.