Tommy - I'm far from an expert, but last spring my wife and I were in Italy for three weeks and traveled from Rome to Venice to Florence and back to Rome during that time -- all via train - as detailed in my OGR post on riding the high speed Italian trains from last June (can't seem to figure out how to link it...).
We booked the Italo portions of our trip from the States before we left - directly on their website, which has an English version. Trenitalia - the other rail service - also has a US version of their website and it appears they have a direct link to Bern via Milan on something called Eurocity - which, from what I can tell quickly, is perhaps something they run. Here's their site: https://us.trenitalia.it-inter...pEAMYASAAEgKcz_D_BwE
I have not used this, but it seems to be a direct link to Trenitalia -- as opposed to a number of ticket sites, like trainline, etc. We had good luck with direct booking. We did use trainline (which I think is based in the UK) for some UK train travel a couple of years ago with no problems.
What we found is that on the high speed trains (both Italo and Trenitalia) there are various versions of premium class tickets which if you look carefully are not all that expensive. The trip from Rome to Bern, via Milan, looks to be around 6 hours plus ~30 min in Milan, so the comfort and meals in premium class might be worth it. I don't know where the line from Milan to Bern goes exactly, but I'll bet the scenery must be spectacular - Italian and Swiss Alps, etc.