>>2029680pic related. this is what the different systems look like. the whole freewheel cluster unscrews from the hub and the spinny, clicky part is inside the gear cluster. the cassette's little cog unscrews from the very end of the cluster, then all the cogs slide off of the hub and the spinny part is inside the hub. the fittings for the tools are different.
once we're for sure its a freewheel or cassette, shopping around is easy. but unless you want to also buy the tool for gear removal and installation, and possibly a big wrench to turn it, you're going to have to take it to a shop anyway. the shop may sell you a wheel and installation for a good price, who knows?
but if you do want to buy the tool and diy, that's fine. there's different freewheel tools. I believe all cassette tools are the same but you have a Shimano hub so buy compatible with Shimano, which is most common anyway.
velomine is known for cheap wheels. this model is currently their only 20" that takes a 7 speed freewheel .
https://velomine.com/products/wheel-master-rear-bicycle-wheel-20-x-1-75-36h-steel-bolt-on-silverso look around for that price range. amazon is fine for what you need, too; or in the US, I like
modernbike.comfigure out cassette or freewheel, then anything 20" that takes 7 speed (the sizing is the same for 5/6/7 so it'll usually be listed like that) and it'll say 135 Over Locknut Distance (O.L.D.) but as long as it says 7 speed, it's good. 135 OLD could be built for 8/9/10 cassettes. I don't think freewheels go that high but anyway, go by the speed listing
