I agree that indexed shifting adds nothing, and second the notion that you clean it up and adjust it and ride it as is. If it was good enough for your mom in the 70's, it's good enough now.

However, if you want to upgrade, I'd start with aluminum wheels. A brand new pair can be had for $80 or less. If you want indexed gears, you can get Sunrace stem shifters for about $21 and the matching derailleurs for another $15. Add a modern chain and modern freewheel or cassette --modern stuff may shift a little better than the old stuff-- and ... please stop!

Seriously, think it through before you start; you can get rear wheel that reuses the same freewheel as you have, or one that takes a cassette with seven or eight cogs. These will involve spreading the frame out wider, which is not a big deal. You have 27" wheels now, but 700c (which is a little smaller) may fit (can you lower the brake shoes 4 mm?).

I think Auchen's point was, if you want a modern bike, get a modern bike. If you want an old one, you already have it. And I tend to agree.