Rail Passes?

We have been in some of those areas and have bought point to point tickets. If you can plan ahead and buy the next leg ticket 72 hours in advance you will get a good discount. Czech Republic, Austria, and Hungary tickets will be less than Lucerne. Don't get a pass unless you want it purely for the convenience of getting on and off wherever you want. The point to point tickets will be a big savings.

"Is it best to get single tickets to each city or a pass that covers all?" Depends on a lot of things, mostly if you know how to use the websites to get advance purchase, discounted single tickets.

Something else to keep in mind (and research) is that sometimes travel by bus will be less expensive. Two years ago my husband and I went from Budapest to Vienna on Orangeways Bus (Orangeways.com) and it was quite a bit cheaper than the train. These buses, by the way, are quite comfortable and have a bathroom and a bus attendant that came around and offered complimentary coffee drinks. Might be worth checking out.

If you post your route, people can tell which web sites to look at, whether advance purchase tickets will save you much money, and how far in advance the tickets need to be bought. For train schedules, look at http://reiseauskunft.bahn.de/bin/query.exe/en. This site will not have prices for trains that do not start or end in Germany (i.e., all of yours), but it is easy to use for schedule information. For pricing and purchasing, use the rail sites of the countries of origin of each trip: Czech Republic: http://www.cd.cz/en/default.htm Hungary: http://www.mav.hu/english/index.php
Austria: http://www.oebb.at/en/index.jsp The big advantage of a pass is that you do not need to pick a specific train in advance. I do not believe that any of your countries require train reservations on these routes, so you could just walk up and get on any train. But, a pass covering four countries (plus any you have to transit - I haven't looked at a map) could be costly. If these are the only trips you're taking, a pass may not pay off, even compared with last minute tickets.

Like German Rail, Austrian Rail gives prices and sells tickets online for some connections that start/end outside Austria but have the other end in Austria. I found connections between Prague and Vienna for sale online for as low as €29 per person (full fare is €62,60/p). Budapest is closer to Vienna. You can get Vienna to Budapest, full fare, for €36/p, or as low as €19/p with advance purchase online. It was difficult to get good prices to Luzern in the near future, but for September, almost three months in advance, I saw Vienna to Luzern for €39/p.