Reddit mentions of SHIMANO Tourney 7Spd 14-28T Freewheel

Your freewheel is broken, you need a new one like: https://www.amazon.com/Shimano-Tourney-7Spd-14-28T-Freewheel/dp/B003B8JYPU/

If you want to do the repair yourself you're going to also need a freewheel tool like: https://www.amazon.com/Park-Tool-Freewheel-Remover-Freewheels/dp/B001B6RGXG

And some grease when you thread the new one onto the wheel.

Here's a good guide on how to do that: https://www.parktool.com/blog/repair-help/freewheel-removal-and-installation

The key compatibility items:

• Rim diameter: 700c (ERTRO 622mm)
  
• Rim width: At least 23mm or so. You can just match the front rim.
  
• Hub spacing: Probably 135mm but you can measure. See this picture.
  
• Hub compatibility: Thread-on freewheel is easiest. See this picture for the difference between a cassette and a freewheel.
  
• Freewheel: I think you can go up to about 28t or so. This freewheel would work but you might find a wheel with the freewheel installed.
  
  Edit: Check the specifics (especially the rear spacing) but this wheel might be what you need.
  
  If you get that wheel you will still need the freewheel, which should spin right on, as well as a tire (you have the specs), a tube, and don't forget the rim strip.

The cheap wal mart bikes can be okay as long as you aren't looking to ride offroad, and replace the shifters/rear mechanicals with proper parts.

That doesn't meant spending a ton of money, the older 7 speed shimano freewheels can be found on Amazon for $15 or so, and so can the derailleurs.

The problem with the freewheels on the walmart bikes is you often see they'll be "slick shift by DNP" freewheels which have a reputation for lasting for about 50 miles then dying. I think I've read they're also made from old shimano tooling, which is worn out so they never shift right no matter how much you adjust them.

As everyone already said, you need a new freewheel.

I cant tell if thats a 6 or 7 speed freewheel, but here is the 6 speed and
here is the 7 speed part you want.

and you need this to get it off.

There are a million freewheel replacement videos on youtube that you can watch to see how its done. Once you get the old one off ( and it will be a bitch to get off... most freewheels are) popping on the new one is stupid easy.

https://www.amazon.com/Shimano-7-Speed-Tourney-Freewheel-Replacement/dp/B003B8JYPU?psc=1

it uses this in the rear so i guess its a freewheel?

Good news! I only need one wheel, the rear:
http://i.imgur.com/Wi2krBs.jpg

So I need a cassette and a wheel. Haven't taken the bike out of my car yet or done any clean up, but I'm probably going to pick up a new saddle as well. That or reupholster the one thats there (wouldn't be my first time reupholstering, though in the past the seats were from cars!)

The front wheel is good, the tire is flat. I'll check tomorrow to see if the tube is any good, or if I need to replace it.

As far as the rear wheel, any reason I shouldn't order something like this wheel paired with this freewheel cassette?

Cheap, but seems to have worked out for like trail riding for others (which is about as hardcore as I'll get).

Thanks again for your help so far!