As this is your first trip, you'll have a bit of a "learning curve". A few thoughts.....
I agree with a previous comment about staying in the Lauterbrunnen Valley rather than Interlaken. Mürren is a very popular location for many on this forum, so it's a good choice. I normally stay in Lauterbrunnen as it's the transportation hub for the valley and very convenient for getting to both sides of the valley. Depending on what sights you want to see there, you might consider getting one of the Swiss rail passes, such as the Half Fare Card. You'll have to do the number crunching to decide which one is the most cost effective for your situation. You may find this helpful - https://www.ricksteves.com/travel-tips/transportation/trains/switzerland-rail-passes .
For rail travel in Italy, there are some potentially expensive caveats to be aware of. The express trains (Freccia, Intercity, etc.) have compulsory seat reservations which as specific to train, date and departure time. You can only use the one train listed on the ticket. If you board another train, even by mistake, you'll likely face hefty fines which will be collected on the spot. If travelling on the slower Regionale trains with tickets purchased locally at a kiosk or ticket office, these must be validated prior to boarding the train on the day of travel, or again hefty fines. The same also applies to using the Metro or buses in various cities, and the Vaporetti in Venice.
Yes, you can easily buy tickets online prior to leaving home, either using the various rail company websites (SBB, Trenitalia or Italo in Italy) or a reseller such as Trainline. The resellers may not be able to provide tickets for every country, so the rail company websites would be better.
You can save money in Italy with advance purchase Trenitalia Super Economy or Economy tickets for the express (high speed) trains, such as the Freccia trains. However there are only a limited number of those for each departure. When the Super Economy are sold out, Economy are the next cheapest. When those are sold out, Base fare (full price) is the only option. The discount tickets come with restrictions, and in some cases are non-refundable and non-changeable once purchased. If you buy discounted tickets in Italy, you must be willing to commit to a specific train departure.
If travelling on high speed routes in Italy, you can also consider Italo Treno, which is a competitor to Trenitalia. You can also buy advance tickets on their website.
You might want to do some research on the aspect of changing trains in Europe, as that will be necessary during your trip. You might find this helpful - https://www.youtube.com/watch?v=VBLrjKFP9Xw .
Good luck with your planning!
