For Rome to Lausanne, you will need to take the Frecciarossa to Milan, then a Eurocity to Lausanne.

Reservations cost €13 per person on the Frecciarossa, and are widely available. You can buy them on the ÖBB website. Add the Eurail discount where it says ‘1 x adult’, and then search for ‘one way tickets and day tickets’.

https://shop.oebbtickets.at/en/ticket

The Eurocity trains from Milan to Switzerland cost €11 in second class, or €13 in first class (only if you have a first class pass!) and are also available from ÖBB.

1.

First you need to decide if you’d like to go by plane or by train. If going by plane, you won’t buy a rail pass (of course) because it’s for (it’s in the name) rail travel. To get an idea: Roma to Lausanne is about seven hours by train and needs at least one change of trains. To verify flight times and prices for your given date, you may use any booking tool.   

2.

If you’d like to buy a rail pass, you need to decide if you’d like to go 1st or 2nd class. 1st class is (slightly) more comfortable, but of course more expensive, too. Be aware to buy reservations for the same class of travel you did buy your pass.

3.

If you’d like a (printed) paper pass, you may choose this when buying your pass (possible here at the website: Eurail Passes | Popular Rail Passes in Europe | Eurail.com). But it will cost you more (you’ll get a paper pass by postal delivery) and it will take time (allow a month for deilivery to the United States; you’ll get a delivery date before checkout). It’s not possible to buy a mobile pass and print it. You may read here how to use a paper pass: How to use your Eurail paper Pass | Everything you need to know | Eurail.com. You don’t need any app to use a paper pass.

4.

If you do buy a railpass, this railpass is your ticket. And now it depends on country and train type, if you do need an additional reservation and / or an additional supplement. For Frecciarossa high speed trains in Italy and for EC trains between Italy and Switzerland, this will cost you 13 € per person and train (if booked with the link at 5.). You may save money by using slower trains and connections with more changes - normally an overseas tourist doesn’t ike to do this but you should know that it’s possible, too. Regarding a trip from Roma to Lausanne in two months (or following trips): There’s a big timetable change in Europe 12/10. Not all trains that will run are already in the system. To verify a train timetable, you may use a date before december 10th and you’ll see what a “normal” train connection does look like. But many trains are already in the system even for later dates. 

5.

You may already book your reservation here: Tickets & Services (oebbtickets.at). You need to book reservations from Roma (Termini is the main station) to Milano Centrale first, then Milano Centrale to Lausanne (allow enough time to change between trains and have a look at the correct class of travel, 2nd or 1st). You need to add the number of persons travelling and you need to add “Interrail / Eurail - Globalpass” as discount. Then you need to tap “find services” and then “one-way tickets and day tickets”. If you won’t get a price of 13 € per person and train, you did something wrong. There’s no need to hold your railpass already, you may also reserve before getting it. Two problems: You are only allowed to choose a maximum of six persons per booking and you can’t choose a specific place - this means it may be true that your group isn’t placed together (especially the persons in the 2nd booking). To avoid this, you may call ÖBB and try to buy by phone. Online, you’ll get a PDF by mail that you may print.

Great explanations above ! I'd recommend reading this link too : https://www.seat61.com/how-to-use-a-eurail-pass.htm

Do not hesitate to ask further questions :)