Going to 7/8 speed cassette isn't a big deal- you can use your original crank, derailleurs, everything. There are some gotchas, though...

1) Your dropout spacing is probably 126 mm, and would need to be spread. Some people can muscle in a 130mm wheel and call it good. Best to cold-set it, though.
2) Is your current wheelset 27" or 700c? If you want to change from 27" to 700c, then the brakes may need to be replaced or modified (sometimes there's enough adjustment room to fit, and sometimes it's close enough that the slot in the brake arms can be filed a couple of mm to fit).

If you want to go to 9 or 10 speed, then in addition you'll need a new crank, probably new rear der, likely new front der, new shifters and cables. At that point it's probably not worth it- you can get a new bike for nearly the same price as the upgrade components.