Be aware that the Amalfi coast is super-expensive, particularly taxis as they are all controlled by a cartel. There are many other bits of the Italian coast line that are just as lovely and much cheaper. If you go further north, it's easier to move around, as well, because the distances between the northern cities are less, and they are well-connected by train and road. Italy has an excellent train system.
If you have to go in the summer, good options could be:
Stay on the coast just east of Genoa. Nice beaches and day trips to Genoa and Milan. Florence is a bit far for a day trip, but you could do it as you arrive or leave - spend a couple of nights there then fly out of Pisa.
As above but stay on one of the lakes.
Stay on the east coast, near Bologna. Day trips to Bologna, Ravenna and Venice - or maybe stay 1-2 nights in Venice - if you stay on the mainland, it's much cheaper. I know people bang on about it being smelly in summer, but I have been there many times in July and August and it's still lovely, albeit crowded.
Stay in Trieste. Day trips to Venice, Slovenia, and Croatia. You can reach 3 countries in half an hour.