How much is the pass? Rail passes are almost never a good deal. Worse since you mention Italy you have to pay for seat booking. IIRC €10 per seat. €10 may not sound like much but for many trips it's more than just buying the ticket in advance. For example Rome to Florence or Rome to Naples if bought in advance can be less than €10 on the high speed train.That means just the seat reservation fee the pass forces you to buy will cost more than the ticket could.

The way to save money is to minimize your movements. The more you bounce around the more you'll spend on trains ,planes or buses.

From the sounds of it you only have one definite stop in Nantes. The simple thing might be to fly into London. Visit. Train to Paris. Train to Nantes. Fly to Rome. From this point it depends on your interests.

I agree that rail passes almost never save you money. But you really do need to hunker down and learn about the various rail systems in Europe and do the math. It takes time, but if you're serious about saving money will be worth it. Start by studying www.seat61.com very carefully.

Maybe also consider Flixbus in your planning. it's a popular low-cost bus service.
Also, check-out events and festivals in the cities you are planning to visit. For example if you don't plan on going to Sziget in Budapest, better to avoid those dates as cheap accommodation will be harder to find.

For the moment, I will just say that I find Budapest much better than overrated Prague. Carry on.

I'm thinking 1st City -> Nantes -> French Alps = 10 to 14 days depending on your pace. The last week can then be used on Italy. You'll have to play around with a spreadsheet to assign activities and days to a specific place before really knowing how many destinations you can fit based on what you're hoping to see/do. Talk through expectations with the entire group.
Lisbon as your first city should be cheaper than London and Paris, but it may cost more to fly there, and I don't know what your interests are to recommend one over another.
French, Italian, and Austrian Alps should be cheaper than Swiss Alps. Switzerland is the most expensive country I've visited, even more than Norway and Denmark, and if you're trying to save money I would steer clear of it.
Bus/trains/airfare are fairly affordable around Europe if you book in advance. But they'll add up if you hop around too much. Traveling also eats up half to a full day from city to city, more if you're going to and from the mountains (like the Alps).
Look at hostels to save on accommodation. Many hostels these days have rooms with ensuite bathrooms. These cost more than the shared bath type but less than a standard hotel. Airbnb if you must, but take care that you rent a legal unit.
Have fun, and I hope you guys stay friends ;-)

Rail passes can be good idea IF you intend to do a lot of rail travel, including a lot of long distance. I'm not up on the latest with the Eurailpass and reservations/fees, so not sure if that has changed things, but if you want to be moving around by rail every few days all over, I think it can be a good idea. And I don't think a 10 euro seat fee changes that. And making dec isions on the fly, when rail tickets can be very expensive for long distance trains. I don't even know if that covers the CR and Hungary, though.

Pal used to push a book/website giving best deals advice to students, what was it called "go..."?

Two general comments:
One -- You face the danger of too much togetherness. Trying to find activities and even destinations that everyone will share can offend different tastes and interests, to the point of losing a friend for life. Sit down now and talk about separate and individual daytime activities.Then you can look forward to sharing your adventures at the end of the day.

Two: On-line travel agencies are useful to stitch together your airline itinerary. Trying to sniff out the "cheapest" airports yourself is far too daunting. Anyhow, except for those travelling on points, airport charges are built into the ticket price. Let an on-line agency show the possible combinations, with no commitment to buy. I like maxtrix.itasoftware.com in the unscientific belief that it covers more possibilities (none will have everything.) It's clunky to use and it doesn't sell tickets but the info can you lead to the airlines that suit you. Buy combination tickets as complete itineraries. And buy as soon as possible; plane and train prices only go up as departure approaches. The days of last-minute bargains on unused tickets are mostly gone so buy directly from the airline if possible. Same for long-distance trains. The single most helpful website, once you are in Europe, is this: www.seat61.com

Sorry, but the old idea of wandering Europe free as the breeze could break your budget. Last minute usually comes at a premium rather than a discount.

If you can commit to at least your trans-Atlantic dates, you may save considerable money and time by flying into one European city and home from another. These must be arranged through a multi-destination search function.

PS: I'm assuming you have already looked at the many suggestions on this site already, through the Destinations menu at the top of the page.