Your frame and wheel wheel are likely to be 120 mm between the rear dropout faces and hub locknuts. 6/7-speed hubs and frames are 126 mm and current 8/9/10-speed hubs and frames are 130 mm.
Assuming it is 120 mm, you will have to add spacers to the drive side of the hub to change the locknut spacing so it will fit a 6 or 7-speed freewheel. Your current axle may be usable but the axle protrusion into the dropouts will be reduced from 5.5 mm to 2.5 mm on each side. You will also have to spread (cold-set) the frame wider to accomodate the wider hub and you will have to redish the rim.
You cannnot fit a cassette to your current hub. You could purchase a complete 6/7-speed freehub wheel that would allow you to use a cassette. It would avoid all of the respacing and redishing but still require widening the frame.
A