roccobike

Mr, Grumpy, OK, now I know what you've got. You have a 6 speed freewheel, not a cassette. You can go with a 8 speed freewheel. I've never worked with one, but I've been told they are available. If you are on the heavy side, you might want to keep in mind that one of the reasons freehubs replaced freewheels is that axles bend more easily with freewheels. An 8 speed freewheel is really pushing the envelope. So if you are up there in weight the freewheel is not a good option. You have 27inch rims on that bike. To convert to a cassette/freehub you'll have to buy a new 27" rear wheel with a 8 speed freehub. You can convert to 700C, but that has its own set of problems with brakes reaching the smaller 700c rim and your front wheel won't match your rear wheel.
Changing an older 27" wheeled bike to index shifting using barcons is not an inexpensive option.
I recommend you look on ebay for six speed Shimano, stem shifters. If you use those, all you'll need is a new Shimano compatible, index compatible freewheel and a new chain. Any bike shop that works on older bikes will have the two latter items. The total cost for making that change should be under $60.