10 day trip itinerary?

It's a bit difficult to provide a meaningful suggestion if we do not know more about your interests.
Do you like the sea? Museums? Roman ruins? Strolling around old villages?

To give you an idea, 10 days is perfect to cover Umbria + Rome, or the Naples area + Rome, but these are very different suggestions.

Venice! I'd say fly into Venice for 3 nights, then take the train to Rome. Book flights as "MultiCity" - best way to go to Europe.

We also love Sorrento - easy train from Rome to Naples to Sorrento. You could do a day trip to Amalfi Coast with sharedtours.com

We also love Herculaneum (easier and just as good as Pompeii to us) - an easy train trip from Sorrento and the RS book tells you exactly how to do it.

With only 10 days, less is more. I'd say fly into Venice, then train to Florence (I know you've been, but you could do 2 day trips with WalkAboutFlorence into Tuscany) - we love their BEST OF TUSCANY TOUR and their CHIANTI WINE & FOOD SAFARI TOUR.

Then take a train to Rome for the last couple of days.

We really need more info - will you have a car, or only Train? We always just do a train, but still manage fine.

9 nights or 10 nights?
When in October?
Are flights set and not changeable?

We spent 9 nights in Sorrento/Amalfi area like this:

Sorrento 4 -visited Pompeii and Capri ( Naples possible)
Amalfi/Ravello 3 nights- visited Positano
Salerno 2- nights- visited Paestum

We began that trip in Rome and ended in Venice

You could head right to Sorrento or Salerno on arrival then end trip with nights in Rome before departure

Umbria would also be nice-Orvieto, Assisi, Spello, Gubbio, Todi…

More info on your interests and travel experience would help

Other than Florence have you been in Italy before?

Do you have the flexibility to fly home from somewhere other than Rome?

Depending when 'in October' you are going the weather may be a consideration in certain areas.

You want to see every museum/historic location possible, or sit in a villa sipping wine?

A few more details from you will help folk here give you useful answers. (The difference between a guided munition and a dumb bomb.)

We are thinking mid-October. We would prefer flying into Rome because it is direct. We are flying to Amsterdam from Italy and back to states. We will not have a car. We are interested in the cities, but want to see Cinque Terre and Amalfi too. We are very flexible. Thanks!

10 days; sleeping in at least 3 different places ; one day will be taken up with going between "Amalfi" and CT

You are looking at a lot of 'traveling' and not a lot of 'being there'.

You really need to choose 'Amalfi' OR CT. Trying to do both in a 10 day trip is masochistic.

With Rome as a starting/ending point, I would strongly lean towards Amalfi Coast.
Plus, there is a lot more to do around Amalfi Coast than around Cinque Terre: Pompeii, Paestum, Capri, Naples even (a bit far for a day trip though)...

At least tell us how long you plan to spend in Rome--that's the low hanging fruit and easy to sort out. Knowing how many days you actually have to work with helps tons.
Are you avid hikers? Cinque Terre would be my choice if you have limited time. While there is plenty to see in its vicinity, there are time constraints with either choice and there is a lot to recommend the Amalfi coast area as its own ten-day trip.

The first time we went we did Venice, Rome, Positano. It was perfect.

Where are you flying out of or don’t you know yet?