This was mentioned already, but arm motions aside, the type of cassette/freewheel the bike is equipped with does matter quite a bit. A modern, Hyperglide-style gear cluster makes regular old friction shifting much faster and smoother. There seems to be a lot less 'hunting around' for the gear, compared to older style (Suntour, etc) freewheels. You typically don't have to 'overshift' as has been described.

Point being, if you're having trouble shifting on an old bike with friction DT shifters, a new cassette or freewheel along with a new chain should make things much easier (plus, it's just good regular maintenance on an old bike).