Well, first of all it makes sense to remind you that with journeys like Barcelona to Paris, Paris to Berlin or Berlin to Rome using train usually not the cheapest option, simply because low-cost aviation nowadays can get you from one place to another cheaper (sometimes even much cheaper).
However if you specifically intend to use trains, then we can give you a number of hints where to look and for what:
Barcelona to Paris: only one direct overnight train - fancy TrenHotel Elipsos train (primarily used by tourists) dep.19:55 from Barcelona Estació de França. Tickets can be purchased either via
Spanish Railways (RENFE) website or via
Elipsos website (company operator). There are some discounted fares, but they aren't very cheap, so in any case it will difficult to count on budget price. Also available daytime options with high-speed train via Figueres
Paris to Berlin: again only one direct overnight train (CNL 455). Tickets (including discounted ones) can be purchased either via
French Railways (SNCF) website or
German Railways (DB) website. Also available daytime options with high-speed TGV/ICE trains & one change
Berlin to Rome: no direct train here (neither daytime, nor overnight). Nearest station form which you can catch direct overnight train to Rome is Munich, so you can take high-speed ICE train from Berlin to Munich and change there (again DB website mentioned above will help you with schedule & tickets). Or, as an alternative, you can also catch direct train to Rome from Vienna. Journey from Berlin to Vienna is a longer one, but Vienna is really nice city, so if you have time you might consider a stop there on your way from Berlin to Rome
p.s. usually ticket sale doesn't open way in advance (for instance DB starts ticket sale 92 days before departure), so you might not be able to purchase all the tickets for November right, don't worry about that, just check closer dates and will give you roughly the same picture
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