First trip to Europe. First trip to Italy. Getting to and from various destinations.

Land in Rome Fiumicino at 1pm. Best way to get to our hotel in central Rome at Palazzo Manfredi? (take train or hire shuttle from the hotel?)

Why not hire the shuttle from the hotel?

Cortona. How to get there?

Take the train from Rome to the stop Camucia-Cortona. Then take a bus or a taxi to town.

Once in Cortona, I plan on renting a car for day trips in that area. Best experiences within a 2 hour drive?

Siena, Assissi Orvieto etc.

Our 5th & 6th nights are in Positano. Best ways to get there from Cortona?

This could be a little more complicated: drive all the way there, or just to Naples, or take the train from Cortona to Naples and then at Naples take the Circumvesuviana train to Sorrento. At Sorrento either take a taxi or the bus to Positano

Use this site to book trains in Italy is most user friendly for new travelers
https://www.thetrainline.com/en/train-times/rome-to-camucia-cortona

You asked for the best way from the airport to your hotel. Hands down that is a private car service. When you are tired, jet lagged, and on unfamiliar turf, nothing beats having a driver meet you inside the terminal and zip you to the hotel. We've used a private car service based in Rome a number of times and have only good things to say about them. The company is Rome Cabs, but it is a car service company despite the name. The web site is romeCabs.com. Check it out and see what you think. They are only slightly more costly than a regular taxi. We always email them with our needs and they respond within twenty four hours.

Google Maps is interactive and can show you the route from the airport, or the central train terminal, to the address of your hotel.

Unless you have your heart set on Positano, I suggest spending those two days in the Umbria and Tuscany region. There are so many interesting places to see. Florence is only a 2 hour train ride. Siena is only an hour drive, plus Assisi, Orvieto are close. Positano, while scenic, IMO doesn't offer much else. Enjoy your trip.

Our 5th & 6th nights are in Positano. Best ways to get there from
Cortona?

Positano is a long way from Cortona for just 2 nights so you will spend quite a lot of time in transit for just one full day for the coast + important archeological sites somewhat nearby. Personally, I'd save this region for a future trip when you can give it more time.

Also, your math seems a little off? If you are spending your arrival night in Rome (night #1) and the next 4 in Cortona (nights #2-5) you would be spending nights #6-7 in Positano, and one additional night in Rome (#8) prior to departure.

It looks as though you're only spending arrival and pre-departure nights in Rome? Have you previously been to Rome? If not, I'd consider adding the 2 Positano nights to Rome instead. 4 locations over an 8 night/7.5 day trip is a lot of packing/unpacking and time spent moving around, especially considering the distance between Cortona and Positano and complexity getting around the coast during high season.

I agree with some of the previous posters. I would use the 2 nights allocated to Positano to Rome. We did the drive last summer from Cortona to Sorrento (so closer than Positano) and it was a solid 5 hour drive. If you are taking the trains, you would need to add at least an hour or two to that number. That would be almost one whole day spent traveling. For us, though, we had 7 nights in Cortona and 4 nights in Sorrento so the long drive was worth it. You would also be back tracking a bit from Rome to Cortona to Positano.
Cortona is lovely! You will want to spend time just checking out the town. Try Snoopy Gelato in Cortona! Our family agreed that it was the best gelato we had during our whole trip. Some other towns to see would be Assisi, Montalcino, Montepulciano or Deruta. I might leave Orvieto for another time or you could stop by Orvieto on your way to/from Rome.

From airport to hotel, www.rome-limousine.com will be less expensive than having a hotel order a car for you. Not much more than the taxis which have a fixed rate from FCO.
How to get to Cortona? Rent a car at FCO and drive there.
You will be on the far eastern side of Tuscany and can also explore parts of Umbria from Cortona.
Perhaps you can fly from Florence to Naples? Then hire a transfer to Positano. That is a lot of traveling for only two nights in a very heavily touristed area. Keep your rental car and drive from Cortona to Naples airport to drop your car off. The drive is not difficult at all. Then hire a driver to take you to Positano or take the airport bus direct to Sorrento, then a taxi, private driver or bus to Positano.
I recommend you spend more time in Tuscany, Florence perhaps. Or in the western parts of the Tuscan region near Siena.

A handy site I use a lot for planning transportation options between any two locations, I.e. two cities, a hotel to an activity, etc. is www.rome2rio.com. I like that it will give you the options down the left column and a map with the names of train stations or bus stops identified in the Italian language.

Some of the train actual times may be off, but there’s a link to pull up the train website, too.

Enjoy your time in Italy!

...our hotel in central Rome at Palazzo Manfredi?..... Have you looked at a map of Rome? Palazzo Manfredi is not in central Rome (IMO). It is a bit of an inconvenient location. Somewhat convenient to the Colosseo but a bit of a hike to everything else. To get to that location I would take a taxi from the airport. The rate is fixed a 48 E within the old city walls. Don't think you would save much by taking the train to Termini and then a taxi. AND I would double check with the driver at the airport to make sure that your hotel is within the fixed rate zone or it will get expensive in a hurry.

I actually think your hotel is in a good location so don't panic. You have the coliseum, then the forum, and across the river is Travestere. There is a bus line/stop near your hotel that goes to Travestere if you are not big walkers. I've stayed closer to Termini, which I had no issue with, but I think your location is actually a little better than mine, and I thought mine was fine. I could be wrong and maybe Frank will elaborate, as Frank knows a lot and I usually agree with him. On the other hand, it is not near the Vatican, which I could care less about, but others might want that to be their focus, so in that case this is less convenient. However, no hotel is near everything in Rome. IMO. Regarding transportation, if you are doing an overnight flight, then take a taxi, hired car, or the hotel shuttle. Otherwise, play it by ear on your arrival and see how you feel, IMO. You also can always walk to the hotel, something I'd be apt to do although it might not be a particularly pretty walk...busy street and I can't recall what it's like.

Thanks for the input everybody. This all started as a trip we won in a raffle for 5 nights in Tuscany (at Cortona Resort) and flights. Wanting to spend some time along the coast, spending a few nights in Positano became a necessity for the beautiful coastline. I through in the 2 extra nights in Rome for the convenience of the flights being direct from San Francisco and wanting to see the some of the major sites in Rome. We couldn't extend the trip any further because of responsibilities at home and I'm realizing this will be a busy 8 days. Getting a little bit of the coast, the city and the countryside should give us a good idea for the next trip back where we will want to focus more time. I just started researching Siena and San Gimignano and have convinced myself I'll at least rent a car while in Cortona for day trips to these cities but now trying to decide how difficult it would be to actually rent the car in Rome on our 2nd day when we'll be heading up to Cortona. It looks like it's only a 3 hour drive or so by car so we could spend a little more time in Rome before driving north on our own schedule. So here's the question, with navigation and printed maps, how difficult is it driving in Italy really? I'm 47 and still have good driving skills and eye sight!