I have upgraded 120mm Schwinn Paramounts to 7-speed shifting as the posters above suggest. It is quite easy to spread the dropouts to 126mm by hand and fit a slightly wide hub in. Once you get to 7 speed you can use the Sunrace 7-speed downtube shifters which are commonly available online and work with (almost) any Shimano rear derailleur. I have mine set up with Dura Ace 7700, mainly because it's shiny.
Another option not discussed above is to go with a 120mm "short" cassette hub. These are sold by either Sunxcd or Grand Bois, though they look quite similar and maybe made by the same OEM. This will fit the International rear dropouts without spreading and opens up the whole world of rear cassettes. Once again you can use a modern derailleur and shifters for index shifting, though the spacing will be different. If money is not a constraint, this seems like the best option.