You've received lots of great suggestions, and I have a few comments to add on each of the trips you'll be taking (more on tickets below).....
- to begin with, how are you getting to Bellagio? From Bellagio to Santa Margherita Ligure, the easiest solution is to take the Ferry across to Varenna (about 20 minutes as I recall), then walk up the hill to the rail station, Varenna-Esino. Buy a ticket from the machine but DON'T forget to validate it prior to boarding the train on the day of travel. I believe there's also a travel agency in the station where you can buy a ticket, I Viaggi del Tivano. You'll have to change trains at Milano Centrale so will require two tickets. Be sure to specify your end destination when you buy tickets in Varenna.
- Santa Margherita Ligure to Monterosso - that trip could either be on a Regionale or an Intercity (reservations mandatory), depending on which train and departure time you choose. There could also be one change or a direct train, depending on which departure you choose.
- Monterosso to Lucca - there will be at least one change and the trip could be on Intercity or even Freccia, depending on which one you choose.
- Lucca to Firenze SMN - that's an easy one. The train will usually be a Regionale and it's a short trip.
- Firenze SMN to Bologna Centrale - that's also an easy one. An easy 38 minute trip via Freccia or Italo high speed train.
- Florence to Siena - you can use either train or bus on that route. One point to note is that the station in Siena is at the bottom of the hill, so you'd have to take a taxi up to the town. The bus drops passengers in the centre of town. The bus station in Florence is very close to Firenze SMN rail station.
- Siena to Roma Termini - you can either take slower Regionale trains (which sometimes involves one leg by bus), but probably the simplest method is to take the Regionale train back to Florence and then high speed train to Rome (~90 minutes).
- Roma Termini to Napolli Centrale - absolutely use high speed train, either Freccia or Italo.
Regarding tickets, I normally buy tickets for the premium trains in advance, so that I can take advantage of cheaper Super Economy pricing. I normally buy Regionale tickets using the machines in stations as it's very easy, but with locally purchased Regionale tickets it's essential to validate them before boarding the train on the day of travel (that avoids hefty fines). Regionale tickets are the same price whether they're bought in advance or on the day of travel.
I have on occasion bought ALL tickets in advance using one of the reseller websites such as https://www.thetrainline.com/ (there are others). Trainline may charge a small fee but that's not a big deal for me. Tickets can also be purchased direct on the Trenitalia or Italo Treno websites. I can't remember whether one has to register on the rail company websites? I did have to register on the Trainline website and provide payment information but once that's done, the process is very easy. I'm more comfortable using my phone for things like that so would probably also use the Trenitalia app on future trips. Of course it helps to have access to a roaming data plan as there may not be WiFi available when I want to buy a ticket.
Note that pre-purchased Regionale tickets come pre-validated so it's not necessary to validate them in the machine prior to boarding the train. Also note that the Express trains come with compulsory seat reservations, and those are specific to train, date and departure time. If you board the wrong train even by mistake, you'll be subject to hefty fines which will be collected on the spot! As I recall, the fines are about €50 PP, plus the cost of the reservation. I've seen the train staff give travellers a break and just charge for the reservation, but I've also seen them apply fines regardless of excuses or hard luck stories.
