If you want to visit Pompeii, it makes sense to stay in Sorrento. It has easy train connections to Pompeii, Herculaneum and Naples (where you will likely want to visit the Archeological Museum to see the truly amazing Pompeii artifacts), has ferries to Capri, and is a real city with all the amenities you need. It is also a resort town. That said, there isn't all that much to do in the town itself. But, it makes a good base.
If you are interested in ruins, put Paestum on the list, as well. It is a pre-Roman Greek town site that is about 60 miles south of Sorrento.
Staying in a town on the Amalfi Coast will greatly limit your travel capabilities. Getting from, say, Positano back to Pompeii would be a long drive in a car (either back along the Amalfi Coast or up through the mountains and Ravello) and I'm not sure what other non-tour options you would have. Positano or Amalfi are small tourist towns where finding other amenities, like food markets, might be tough. Things will certainly be more expensive in the Amalfi Coast towns then in Sorrento.
One last point. The Amalfi Coast road is a narrow, 2-lane road used by large tourist buses that act like clogs in a pipe for traffic. That is, a tourist bus is really too big for the road and buses will always have a line of cars behind them. After the cars pass, there is no traffic until the next bus goes by. Driving on this road in a car must be frustrating as most of the time is spent moving in a slow traffic jam. On a bus, you have no control over where you stop and you might end up on the wrong (left) side of the bus where the views are not so good.
In addition to taking a bus down the coast, I have ridden a bicycle on it. While you need some fitness to do this (though, renting an electric bike would help solve this problem), it is far and away the best way to see the Amalfi Coast. While the road is dangerous in a car or bus, it is completely safe on a bike. When the buses go by (you can hear them long before they pass), you pull off the road, wait for all the cars to pass, and then continue riding in the quiet. Driving you will SEE the Amalfi Coast. Riding a bike, you will EXPERIENCE the Amalfi Coast. Here is a link to my biking journal for
the day I rode from Sorrento to Salerno, along the Amalfi Coast. This will give you a good idea of what riding down the coast is like.
No matter how long the hill, if you keep pedaling you'll eventually get up to the top.
