Make sure you have your terminology correct.

See Sheldon Brown's page on Freewheels vs Freehubs/Cassettes

You can buy Freewheels in 1, 3, 5, 6, 7, or 8 speed. 9?
You can buy Cassettes in 6 to 11 speed options.

Upgrading from a 6 to 7 speed freewheel is easy enough. Just unscrew one, and screw the new one on. Fiddle with the spacers so you get enough clearance, and perhaps re-dish your wheel to make sure the rim is properly centered.

Changing from a freewheel to a cassette is a much more labor intensive process requiring the replacement of the entire hub, and probably the spokes (or, just find a cheap wheel to replace yours).

Many (but not all) of the Shimano brand freehubs are interchangeable, and can be upgraded or downgraded as needed.

There are lots of axles and skewers available, although virtually all skewers are metric, some of the solid axles are "standard", so you're best off finding an axle with the proper cones and spacers. But, overall the change isn't too bad, and is a good opportunity to clean and grease the bearings, as well as replace bent axles.