The difference is that 1st class seats are larger and more comfortable than those in 2nd class. There are no other added benefits in regional trains, because as far as I know seats aren't numbered and many times people from the 2nd class take a seat in 1st class anyway... also, 1st class is not always present in common trains. In Frecce (Frecciargento, Frecciarossa), on the other hand, all seats are numbered. The biggest difference, really, is in the price!
When a train is about to leave or arrive at the station, you'll often hear announcements such as: "Il treno regionale [train ID] delle ore [hours] per [destination] è in arrivo al binario [number]", "Il treno regionale 21003 delle ore 16:10 per Ancona è in arrivo al binario 3" if it's arriving, or "Il treno regionale 21003 delle ore 16:10 proveniente da Ancona è in partenza dal binario 3" if it's leaving (I'm making the numbers up here). Piattaforma is seldom used in this context because it's the area where you wait for the train to arrive.
Bagni is what we most often use for public bathrooms, such as in "Mi scusi, può dirmi dove sono i bagni?". If you are guest in a house you would use the singular, "Può dirmi dov'è il bagno?". Another synonym for public bathrooms is servizi, from servizi igienici, "Può dirmi dove sono i servizi?". Literally, services. People would understand you with W.C. anyway but it would sound strange to the ear.
I hope this helps!