Where would you recommend staying in Italy for a 4-5 stay that would include convenient attractions, good meals, hiking, and scenery for a family of 6. All are in good health.
Florence. Busses go from Florence to hill towns in Tuscany.
Sorrento. A lot of walking/hiking options. Beautiful city. Easy access to Amalfi coast, Naples, Pompeii.
What time of year?
Will you have a car, not have a car, or could you go either way?
When you say "convenient attractions," what did you have in mind - water parks, Roman ruins, art museums?
I'd visit Venice for a couple of days, and take a train down to Florence--just over 2 hours away.
Rome is a 4-5 day stay to itself.
Get an apartment in Rome. It has everything on your list, especially hiking.