Changing Front Chain Rings & Rear 7 Speed Cassette

I've three in the front & seven gears on the rear wheel.

In the last two years I've replaced (the chain and) a broken rear derailleur with a Shimano:

Tourney 6/7 Speed SIS Index

The front derailleur was missing when I got the frame from a friend, It's a Rocky Mountain btw.
(I haven't looked hard enough yet to find a good FD that fits the tube size...)

In the next phase of rebuilding, I'm looking to replace the Cassette as I can see the teeth are worn down, and the chain jumps something alfull when going over bumps and/or under high torque pedaling.

I want to keep the existing rim, which I bought new around the Holidays.

I've been riding most all my life but never really got too far past keeping air in the tires & the chain good and greasy.

Doing a quick Google search found me this:

https://www.jensonusa.com/Shimano-CS-HG ... d-Cassette

a Shimano 7 speed, but I'm not sure how to determine if it's compatible with what I already have,
that and I'd really like to increase the Granny Gear action of the largest, if not the two largest gears.

I know enough to know I don't know enough...