What other parts might need replacing if I replace an old freewheel gear set for a modern freehub cassette?
You probably have the right drop out width.
Unless it has down tube shifters you have cable routing.
The chain size must match (e.g. cassette number of gears) and chain ring width.
Your existing chain rings are probably too wide.
But I think you would be better off finding a newer used bike with the stuff you want. 

If you stay with the same number of gears and get different gearing (number of teeth) you can get better climbing.