A lot of "racers" of that era were bought as 5-speeds which kept the cost down and were perfectly OK for general use. My own Raleigh Arena bought new was a 5-speed, and I went everywhere on it.
Rather than go 10 speed, I would seriously think about fitting a six speed freewheel to replace the 5 speed . Presumably you've got a 14-17-20-24-28 on there now. I find a 14-16-18-21-24-28 is much nicer to ride with as the gears are better spaced. Won't help with hills though.