If you have 5 in back, then it's a freewheel, not a cassette. You can only go to 6 or 7 with a freewheel. You would need a remover tool to match your brand, and probably a new chain, especially if you go with narrow ultra-7.

To go to a cassette with 7-10 gears, you'd need a different wheel with a cassette hub, then a matching chain.

Pretty much nothing is universal ... things have changed over the years, but some things are more common or standard than others (700 c tires, English threaded bb) .....

This will tell you 99% of everything there is to know about bikes:

http://www.sheldonbrown.com/