Sorrento, Amalfi or Positano

That area can easily fill over a week and make you feel you wished you had more time. I would say that 4 or 5 nights there is the bare minimum you should plan. These are some of the highlights:
Amalfi, Positano, Ravello, Capri, Sorrento, Pompei, Ercolano, Naples, Procida Island, Ischia island, Vesuvius, Paestum, and much more.
Although I love to stay in the Amalfi coast, however if you don't have a car, Sorrento is logistically a better base. You can easily visit everything else from there.

Even with a car in 2 years ago, we stayed in Sorrento (actually, in Piano di Sorrento next door). The B&B where we stayed also runs a cooking school - www.mamicamilla.com. If you're intrigued by the ruins at Pompeii, and if this doesn't add too much hustle & bustle, also consider visiting the ruins at nearby Herculaneum.

Without a car you could spend a night or two in Paestum and the rest of the time in one of the Amalfi Coast towns, or Sorrento. I like Praiano, about halfway between Positano and Amalfi. The Amalfi Coast is more convenient with a car. I can recommend a small hotel with free parking - even the car will have a sea view.

You really don't need a car. When you go from Rome you still got 9 days, right? You can stay one or two nights in Naples. And you don't want a car there, trust me. There are several ways how to get to Sorrento from Naples. IMHO the best is by boat. Have your base in Sorrento and go by Circumvesuviana train to Pompei. By boat to Capri. By bus to Amalfi. Sit on the right hand side. Wonderful scenery. Back by boat, stop at Positano and then by the last boat to Sorrento. Sorrento is absolutely ideal to sit in a cafe outdoor at the main square or at the hotels cafes (outdoor,too) on the edge of the cliff and have a view of the Naples Bay. Sometimes you can even see Naples itself. You are worrying that 9 days could be too much time for that area. 2 days in Naples, 7 days in Sorrento. Day trips: Pompei, Vesuv, Capri, Amalfi coast, some rest in between and there are some gorgeous hikes in the area. Ultimately you will think you don't have enough time there.

Sue, you didn't give the month of the year. As much as I liked Sorrento, neither Sorrento nor Positano can be described as "undiscovered". Amalfi (and Ravello) are nicer, but also heavily populated in (a long) season. Do you care if the beach is pebbled and isn't always free? Unlike CT (which I hear is quieter after 5PM) these towns are all busy at night, too.

We are going end June to July 14. I am thinking 4 days in Sorrento, three in Amalfi. Sorrento we can see everything, then layback in
Amalfi.

Atrani on the Amalfi Coast might be a good place to lay back. I like Sorrento, but it's not the Amalfi Coast.

We stayed in Atrani on our first visit to Amalfi Coast Sept '14 (so hard to think it was last year already!). We used an airbnb apt rental. What was great about it is that it was a 5-10 min walk to Amalfi. We stayed 3 nights...the 'bad' thing was there was no parking in the village as it's basically just one small road thru the middle (unless you are a resident/have a permit) - there was a paid parking lot near the beach, and a very large parking area built into the hills between it and Amalfi - which we discovered on the last day you could actually walk thru instead of walking along the road. Or of course parking along the coast road (can you say scratched vehicle?) Didn't matter for us as we didn't have a car, and it is a really tiny village, so you wouldn't have to drag your bags too far.

I actually would stay again - I kinda like smaller villages (yes, I stayed in Corniglia twice on the Cinque Terre - just because it was the smallest and quietest).

The airbnb guy picked us up at the bus area in Amalfi and drove our luggage and us around the corner and helped us take our stuff to the apt (if you are interested in the apt, LMK and I'll send a link - or I did post it under Italy review section)