Train guidance for a group of 9

It's easier than you think. First, there is no local train from Termini to Trastevere. Take the bus to Largo Argentina and then the tram into Trastevere. You just need to find out from the apartment owner which stop to take.

From Rome to Florence? Book on the Freccia line - it's very fast and very comfortable. And yes, you can pre-book tickets. If you book for a specific time and class of service you can save some real money.

You don't need to pre-book to Orvieto. and back to Rome. It's a regional service. You can get your tickets in advance on line, if that's easier for you, but it isn't necessary. Have fun!

I assume you're looking to take the Michelangelo Express into Termini and then get to Trastevere to your Air BnB. You might want to contact one of the shuttle services or taxi companies with vans to carry 9 people. The Michelangelo Express is relatively expensive, and it may be cheaper to catch a shuttle directly to your accommodations than taking the train in. I think you'll find Trastevere as a great place for restaurants, and it's a very popular place.

Yes, there is a local train from the airport to Roma Trastevere station. Every 15 minutes M-F, 30 minutes Sat-Sun. If it is not close to your airbnb, take the tram or a bus to get closer.

There is also a competing fast rail service operated by Italotreno to Florence. Advance nonrefundable tickets available at a discount.

Florence to Orvieto is mostly by direct Regionale veloce trains that are always the same price at 16.90 EUR pp.
No reservation taken or needed. Get to the platform early and you can all board and sit together.

Ditto Orvieto to Rome.

Are all of your group adults? There are discounts for kids, depending on their age. Also you may be able to get group discounts, at least for the Rome-Florence train.
https://www.trenitalia.com/en/offers/travelling_in_groups.html

what is the 'best' way to understand the local train from Termini to the Trastevere

At the counter ask for 9 tickets for the FL1 train to Trastevere and not for the Leonardo Express train to Termini. It's 8 € pp, one way. Not 10, not 14: the FL1 costs eight Euro per person one way.

If so what service should I use?

There are two companies running high speed trains from Roma to Firenze, use the service with lower prices and better schedules.

Is it possible to pre book trains from Florence to Orvieto? Same question for Orvieto back to Rome for departure on 4/16?

You can.t book local trains, but you can get tickets in advance on the official trenitalia.com Site. Use the real dd/MM/YY date format. Unlike the tickets purchased online, Local trains' tickets purchased at the counter must be time-stamped before getting on. Do not forget ti time-stamp the FL1 tickets.

Same question for Orvieto back to Rome for departure on 4/16?

If your departure is before 5PM (17:00) you need to be in Rome the night of the 15th, or you risk a very unpleasant end of your trip.
You could end adding an extra day and buying some really expensive airline tickets to replace the ones you could not use due to being late.

If you still have the flexibility to do so, you would be better off to go to Florence first and do Rome last.

Ticket lines at Italian rail stations can move incredibly slowly. I wouldn't recommend buying the tickets right before departure, even for the regional trains with fixed ticket prices. That doesn't mean you have to buy the regional tickets before you leave home. I often buy outbound tickets at the train station as I arrive in town. If there are ticket machines (there probably are; I just don't remember), try those. Occasionally folks report encountering European vending machines that don't like their US credit cards, but that has only happened to me once (not in Italy).