Meg,
A Railpass probably won't be the most cost effective choice, but you'll have to do some number crunching to determine the best option. You may find it very helpful to have a look at Rick's Rail website, which provides a rail cost planner, which will help you to get an approximate idea on the costs. You could also have a look at THIS website to get some idea of the best options.
Regarding each of your trips.....
- Rome-Florence: use Freccia or Italo high speed (300 kmH) trains. VERY easy! If you're willing to commit to a specific travel time and date, you can pre-purchase tickets about 3 months in advance and possibly save money. Note that Italo uses different stations in Rome, and therefore Freccia might be the easiest choice. NOTE that each city has more than one rail station, so you'll need to be clear on the details before buying tickets!
- Florence-Venice: Again, Freccia or Italo. Both use the same stations.
- Venice-Munich: Flying to Munich wouldn't necessarily be faster. I'd use a departure from Venezia S.L. at 07:50, arriving Munich Hbf at 14:21 (time 6H:31M, one change at Verona PN, reservations compulsory).
- Munich-Salzburg: A very easy trip via RailJet, as short as 1H:28M. Lee or one of the others can help further with information on ways to get cheaper regional tickets.
- Salzburg-Prague: The easiest trip seems to be a departure from Salzburg Hbf at 10:12, arriving Praha hl.n at 16:21 (time 6H:19M, one change at Linz).
- Prague-Heidelberg: I'd probably use a departure from Praha hl.n BUS station at 09:35, arriving Heidelberg Hbf at 16:48 (time 7H:13M, one change at Mannnheim and a 10 minute/300 M. walk to the rail station).
- Heidelberg-Paris: One of the shortest trips is a departure at 09:13, arriving Paris Est at 12:50 (time 3H:37M, one change at Mannheim). You can get from Paris Est to the area where your hotel is located using Metro. It's sometimes a bit awkward with luggage, but certainly possible (I've done that numerous times).
- Paris-London: EuroStar from Gare du Nord to London St. Pancras is of course the best choice. If you book really early, some cost savings are possible. However, the cheaper tickets are often non-changeable and non-refundable, so choose carefully! You can get further information on that trip using the excellent Man in Seat 61 website.
For researching all your rail trips, the bahn.de (German Rail) website is usually the best and most accurate source of information.
I should stress that I've only had a somewhat cursory look at your proposed Itinerary, so this is just a few brief thoughts on the trips you're planning to make. As April is only a few months away, it might be a good idea to pre-book at least some of your accommodations. If this is your first trip to Europe, it would be prudent to read Europe Through The Back Door prior to your trip, as that has a lot of good information on "how" to travel well in Europe. Use the country-specific Guidebooks to plan lodgings, transportation, sightseeing, etc.
Good luck with your planning!