I would guess that you have a 7 speed FREEWHEEL, and not a cassette. Not that it makes a huge difference (except when replacing it).

Post some photos of the bike, crankset/chainring, and freewheel. One-piece cranksets are common. Apparently the chainrings can be replaced, and aren't too expensive (also a good time to lube the bottom bracket).

But, you could easily end up spending $50 on parts if you're going to a wider range freewheel and a smaller chainring.

Tools?

Count the teeth on the freewheel a bit more precisely. 14T or 15T? Perhaps remove the rear wheel to count them.

DNP Epoch makes an 11-32 freewheel.
https://www.niagaracycle.com/categor...-nickel-plated

The tooth progression isn't uniform, so it is quite a jump from 24T to 32T. "escape gear". But, by dropping a few teeth at the top end, you can go with a smaller chainring and not loose power at the top, while getting easier at the bottom.