You'll find the trains easy to manage. Just keep track of where you are vs. where you need to hop off the train and be near the door with luggage in hand as you approach your station. The whole process will be more pleasant if you aren't burdened with a lot of luggage. (The same will be true of the rest of your trip, as well.)
First, you'll need to get to Florence's Santa Maria Novella station. There may be a public bus that would help you there, or you could arrange for a taxi. Your hotel or landlord should be able to advise.
You can check train schedules and buy tickets on the TrenItalia web site. It requires that you use the Italian spellings of station names, so you'd use Firenze S. M .Novella for Florence. At the moment you will only find schedules up until early December, but the schedules probably won't change a lot before your trip. (Which is when??)
Getting to Vernazza by train will not be a problem. You'll probably need to change trains in La Spezia, and possibly also in Pisa. Some trips are all on regional trains. Those have no reserved seats and there's no advantage to buying your ticket early, though you don't want to wait till just before departure time since there might be a line at the ticket window. Sometimes the ticket-vending machines are not fond of US credit cards, but they are certainly worth trying.
Some of the Florence-Vernazza itineraries include a faster train (Freccia or InterCity) for part of the journey. Those legs require reservations (and you are guaranteed a seat), and the tickets for those legs can be significantly cheaper in percentage terms if you snag one of the promotional-fare tickets shortly after the tickets go on sale. Still, it's a short trip and we're probably only talking about a difference of $10 to $15 per person one way, so I don't think you need to agonize over it. The promotional tickets are non-refundable and non-changeable, so you'll want to be very sure of your timing before making the purchase.
Although taking an express train for part of the trip sounds like a good thing, it will not always save you a significant amount of time, so do pay attention to the actual duration of the trip.
If I understand you correctly, after your visit to the Cinque Terre you plan to travel to Rome (use "Roma ( Tutte Le Stazione )" on TrenItalia) on Sunday, then you are flying back home on Monday. You can certainly make that trip from Vernazza on Sunday; no need to change hotels on Saturday night. This is a longer trip, with more time on one of the fast/costly trains, so you can save a good bit of money here by figuring out your schedule early and buying the tickets early on.
Most trips to Rome involve a change in La Spezia. For this Sunday I see an 8:53 AM departure from Vernazza, arriving in La Spezia at 9:13 AM. There are at least a couple of options at that point. The simplest routing puts you on a fast train for Rome that doesn't leave La Spezia until 10:06 AM, so I wouldn't be concerned about the connection. A faster routing overall--but one requiring two additional transfers and a lot more time on regional trains--has you leaving La Spezia at 9:23 AM. So that would be tighter. As a default TrenItalia didn't show me that 9:23 departure; it tends to focus on trips maximizing the use of the fast trains, even though in this case the fast train gets you into Rome later.
Whatever itinerary you choose, check during your time in the Cinque Terre to be sure there's no rail strike planned.
There are lots of ways to get to the airport, which is where I'd head if I were staying at a hotel hear there. You'll be coming in on a train, so I'd use a train from Rome to the airport. The Leonardo Express runs from Roma Termini station. There's also a cheaper regional train running from Roma Tiburtina station. At least some of your train options will stop at Tiburtina before going on to Termini.