Best mode of transport is Euro trains. It’s fast, well connected and you can see beauty of Europe while travelling. For budget hotels I think @accorgrouphotels is a good choice. @Booking.com has many options for every budget. If possible do visit Verona while in Italy. It’s a small city with Juliet’s house as main attraction.
In paris, stay close to louvre, its cheaper than the region near eiffel tower, must dos alart from main attraction mont parnasse tower(best view of Eiffel tower), Trocadero Plaza, Notre dame cathedral, Arc de triomphe, Moulin rouge Cabaret. Visit louvre on a wednesday/friday, they are open till 9 Pm on these days and so less touristy. You can use HOHO BUS or metro. it is very easy to move around the city. For intercity, since you are going 3 countries, buy a eurail pass. For switzerland as well buy the swiss pass which will be cheaper. it will cover alot of attractions in the pass itself apart from metro and buses. Take the day trip to Pisa early in the morning when in florence, because later the traffic increases and takes longer to reach. Avoid using buses in rome during 9-12 in the morning, rush is at its peak. Again use HOHO buses in rome, they are very good. Some ticket combos include tickets into colosseum and Vatican City, look for them.
Hotel Londra - Firenze for Florence
Hotel Napoleon - Rome I have stayed at both these hotels . They enjoy a central location , close to major attractions and from hop on hop off pickup points . You may go ahead with them
Switzerland - which cities you are planning ?
Ya this is a good idea . Switzerland is a small yet very well connected country . You can easily keep base at one place and then explore the entire country on day trips . Buy 4 day SBB unlimited train pass. It will entitle you to unlimited usage of all public transport in Swiss 9 ( including boat / ferry ) pass will cost 218 CHF per person for 4 days
Paris - Hotel Di Cadran
Extremely close to Eiffer Tower,Champ De Mars,Champ Elliseey,Trocadero,Opera House Florence - Nilhot Venice - Hotel Al Codega Rome - Bettoja Atlantico Hotel Zwitzerland (if you are staying in Zurich),go for Swissotel.Its a little expensive but is class and js closer to Hauptbahnhoff (the city center).