Transportation Advice, please!

Couple of things, Trenitalia is the national rail company in Italy, there are some other private companies that may work for you, but their site is a good starting point. For longer trips, buying tickets well ahead can save you money, if cost is not a big issue, do it there, buying a day or two ahead, or buying your ticket to leave when you arrive. All European schedules are pretty specific about both station name and spelling, so research to make sure you have it right, also be aware that many towns have multiple stations.

Not enough time to comment on everything, but to get you started...

From the train station at the Rome (FCO) airport, you basically have two train choices. One is the Leonardo Express which will take you directly to Termini train station, but you would need to backtrack to the Trastevere. For you however, there is another train the FM1 that is cheaper, and stops at Trastevere Station. From Trastevere Station there is a Tram that heads up ,via Trastevere, across the river, and into Largo Argentina. If that works for your hotel, great, if not, you can always catch a Cab from the station.

Your Rome to Venice leg, you need to determine your departure station, Trastevere is likely not it, or at least directly. It would likely be Roma Termini, but some leave from Roma Tiburtina. On the other end, there is no Venice Central, the Station in Venice Proper is Venezia Santa Lucia, sometimes noted as Venezia S. Lucia. From the station, yes, you can catch the vaporetto to Piazza San Marco.

Skipping the next couple, Florence to Naples is pretty much a straight shot, again, a good discount if purchased ahead. Myself, I would not recommend a car for the Amalfi Coast, especially "in season". Too congested, very tight roads, parking a nightmare. Either Naples or Salerno are good options, but for Raito, Salerno would be handier. A car might be handy for around Salerno and sights away from the Amalfi Coast, but I would take a bus or boat into Positano and towns along the coast.

To get to Fiesole, you can take the Florence city bus #7 from alongside the Museo San Marco, right near the Accademia. Edited to add that you can buy the tickets at news stands and tobacconists', and you need to validate them by punching them in the machine on the bus.

And Florence main station is "Firenze Santa Maria Novella" or Firenze SMN. You need to have the right station names to use the Trenitalia railway ticket site, http://www.trenitalia.com/tcom-en

Venice to Menaggio - take train from Venezia SL to Milano Centrale (impressive and historic station). Take regional train from Milano Centrale to Varenna-Esino, Walk three blocks to ferry landing, take ferry to Menaggio. Enjoy your trip!

That's 3 short blocks, downhill. (of course on return it is 3 short blocks uphill).

Fiesole has never struck me as rural.

Firenze SMN-Salerno 3H45m

Then rental car or taxi 20 min.
Just check that your lodging has parking.

Rome to Trastavere:

Since there are two of you plus luggage consider booking a transfer directly to your hotel see Viator.com

If you want to save money then yes you can take the express Da Vinci train to the main train station Termini. From Termini, you can take the bus or tax. Don't walk it is way to far. Google Maps (directions) will give specifics of which bus to take. The whole trip will take about an hour. The buses pick up right in front of the train station but be careful because one side on the median is one direction and the other is another direction. Just say "Trast-ever-eh" to the driver and he'll point you to the right bus. There is no under-carriage luggage storage so you'll have to put your bag between your legs (this route isn't very crowded). Also, note that the buses do not go into the Trastevere. You'll be dropped off on the other side of the Tiber, walk across the bridge and then take the stairs down. Most of this area is cobblestone so a roller-bag is "idiota" as RS points out in nearly all of his European videos and talks.
Have a great time Trastevere at night is my favorite part of Rome.

Amalfi:
Search the forum for Amalfi there is tons of information on how to get there and what to see along the way. Getting from Amalfi to Rome can take a while. You may have to go back to Naples or Rome the day before your flight home...based on the train schedule and your departure times.

The FM1 train from FCO to Trastevere worked well for me last summer. I opted for a taxi from the Trastevere station to my hotel because of the hotel's location on a back street, the (unfortunate) weight of my 24" suitcase and my concern about wrestling said suitcase onto the tram. When I later saw the tram, I realized that I would not have had a problem using it because it sits close to the ground. I would have had to deal with cobblestones, though.

You say you are a detail person. So am I. Here's a suggestion: Rather than trying to compile all the information you need/want from lots of different sources, you might just want to pick up a copy of the Rick Steves Italy guide book. He's loaded with specific details, advice, suggestions, etc. It will be the smartest money you ever spend.

Also, he answers questions you don't realize you need to be asking.

Happy travels.

Michael beat me to it, but I was also going to suggest/insist that you get Rick Steves Italy. He has all the details of how to get to the places you want to go, including very useful nitty-gritty (such as which side of the street to stand on for a bus in a particular direction, or if the bus stop for a return journey is around the corner from where the arriving bus left you off).

Thank you all for your responses- they've been very helpful! and yes, of course I have the Rick Steves book, and am using to to do much of our planning :) If I understand correctly, this seems to be the best course of action:

Upon arrival at FCO, take the FM1 (don't prebook tickets, becuase flight times are variable and it will make more sense to purchase tickets there). Either take the tram or catch a cab to the hotel in Trastavere.

Rome to Venice, take the train from Roma Termini to Venezia Santa Lucia. Here it will be helpful to prepurchase tickets, due to potential savings, yes?

Venice to Menaggio, take the train from Venzia S Lucia to Milano. This is another where you all might recommend purchasing tickets. Then the regional train to Verenna-Esino, then ferry from Varenna to Menaggio, no need to prebook either of these.

Milan to Florence, take the train and prebook tickets. Then take the City bus #7 into Fiesole...though, to be honest, we might just cab to the Pensione, becuase I might like to have my bearings well in hand when I try to become familiar with local busses.

Florence to Naples via a fast train, to be prebooked. Then...I can take a train from Naples to Sorrento (probook, or no?) and pick up a rental car there so we can drive around for the week that we will be in Raito (we plan to head east to visit relatives, then also north to Caserta). We will likely ferry into Positano, and will probably not want to drive the Amalfi Coast ourselves, right?

Then we can take the train from Sorrento to Naples and Naples to Rome, spend the night in Rome, and fly back the next day.

Again, thank you so much for all of your help in nailing down these logisitics! We are so excited for our anniversary/prebaby trip!!

In case it wasn't clear from earlier input, to get the big savings on fast trains you generally have to buy the tickets way in advance, often shortly after they go on sale. It's not just a matter of getting them a day or two or a week ahead. The savings can be really significant--50% or even more. Buying two tickets, the price difference can add up quickly.

A second way to save money on trains (most practical on not-terribly-long trips) is to book a slower train rather than a Freccia. Sometimes the time difference is pretty minor and the savings are significant, and you avoid the not-insignificant Freccia mandatory-reservation fee. Only you can detemine the relative importance of time vs. money. The advantage of this technique is that you retain some flexibility since you aren't tied to a particular train months in advance. However, during the busiest travel periods (holiday weekends, commuting hours) it can put you on crowded trains, and there's at least the theoretical possibility that you'll have to stand.