Index shifting works fine--- although nothing is stopping you from using friction.

I'd start with a new Deore rear derailer, 12-32 or 13-32 rear cogset or freewheel and new chain. Your bike might not allow the use of an 11 thooth cog. Performance bike has all this stuff for under $50.

You currently have 2 sets of barend shifters (either one will work fine in friction mode with Shimano parts)

You'll need a triple crankset, bottom bracket, and a frount derailer. Hunting down a used crank and matching derailer isn't a bad idea, or get a new Sugino crank and Sora FD. Buy the best bottom bracket you can afford.

The cheap way will cost less than 100 bucks

The right way, with better new parts and custom wheels, could run you $500.

Remember, it's just a bike and have fun!