You might want a car for Lake Balaton, but otherwise take the train. See seat61.com for general train info and bahn.de for schedules.

For Budapest I highly recommend an apartment from this site: http://www.budapestvacationrentals.com/Seven_Very_Special,_Very_Central_Budapest_Vacation_Flats.html - I stayed in Liesel in November and loved it.

Triple rooms can be a challenge, but not impossible. We stayed in Vienna at the wonderful Hotel Kaiserin Elisabeth, in central old town near St Stephans. It is a lovely boutique hotel, but is somewhat expensive. We had their 'family room', which had one double bed and two twins, one bath ensuite. Breakfast was included in their beautiful lobby room.

Also in Vienna, this place is popular:
http://www.pertschy.com/
We had dinner there one night and it looked very nice.

For Prague, try LivingPrague.com It is a wealth of information on lodging and all things Prague. We stayed at the Pension U Lilie, to be in the center of things and to have a room for our family of four, and breakfast included. It was ok. A little tattered.

As far as Jewish sightseeing is concerned, Precious Legacy Tours www.legacytours.net is considered the finest tour company specializing in Judaica and Jewish interest touring - it is the only thing they do and they are the experts and have the connections. Disclaimer, DW worked for a time for the co-founder when he was the person responsible for finding and putting together the Precious Legacy Exhibit of historic Czech Judaica that was a special exhibition at the Smithsonian and other museums around the world. They offer Jewish interest tours in Prague, Vienna and Budapest as well as other places as well as specialized tours, e.g., Synagogues of Bohemia.

Trains between main cities are getting up to western European quality and thus are a great way of getting around - and there is a bargain-priced, as railpasses go, European East Railpass that lets you hop on practically any train anytime - just show up - valid in Austria, where trains can be particularly pricey, Poland, Czech Republic, Slovakia and Hungary - for some great IMO info on trains in Eastern Europe I always highly recommend these sites - www.budgeteuropetravel.com; www.seat61.com; www.ricksteves.com. And for schedules for all European trains I always use the Wundbar IMO German Railways web site www.bahn.de.

Decades of travel on European trains makes me exhort folks to pay more and go first class - not that 2nd class is a cattle car though it may be a times IME but first class much more relaxed - always in my experience empty seats so you can just show up and jump on - bigger seats, easier to stow luggage, etc. and the European East Railpass is especially good priced compared to regular first class fares - a few trips can make a 5-day (flexi to be used at will over a one-month period) pay off - especially in Austria!

I would definitely take the trains. We bought our tickets in advance for Vienna-Prague-Munich, from BudgetEuropeTravel.com. You can call their 800 number and they are extremely knowledgeable and helpful.

Be very careful with taking public transportation in these cities, especially Budapest and Prague. The undercover transit police target tourists and there are heavy fines for not having a valid ticket.

Fun, thanks for the heads up! The joke is that all of us speak just enough Hungarian to be dangerous and could curse out (and have) someone pretty good. The girls and I are laughing that we will have to control ourselves and our "secret" language because obviously in Hungary, we would be understood.

You can also take the hydrofoil down the Danube from Vienna to Budapest - it's about a 4.5 hour ride (about 50% longer than the train) but may be worth it for the scenery and photo ops.

Private guides are generally affordable and easily found for specialty sites. Precious Legacy should work for Prague but if you wander around Josefov, you should be able to find a bunch of brochures. In Budapest, you can get a pile of brochures at the Dohany Utca Synagogue in Budapest for tours of the old Jewish Quarter. You can also pop into any sizable hotel in either city and pull brochures.

You may also want to take something like this: http://www.absolutetours.com/tours/regularly-scheduled-tours/hammer-sickle-tour.html. It's especially good for a Monday when the other museums are closed.

No reason to fear the cops in Budapest or Prague, just get your ticket validated at the machines (for Budapest) and don't mess around.

And this is a good info site for Prague: http://www.livingprague.com/

Drive as little as possible. The public transport is generally good. Prague's most touristed areas are all walkable and relatively close together. Budapest's metro and trolleys connect major sites fairly well. And you can't rent a car in one country and drop it in another without paying a small fortune for a drop charge.

Even better in Budapest, buy a multi-day pass. No need to validate it for every trip, and you can hop on the metro for one or two stops without worrying about the cost.

See: http://www.bkv.hu/en/types_of_tickets_offered_for_tourists/types_of_tickets_offered_for_tourists

I highly recommend a night or two in Cesky Krumlov between Prague and Vienna.

I took a 'day tour' 8-seater van from Vienna to Budapest. They picked me up at my hotel early in the morning, threw my luggage in the back of the van. When we got to Budapest, we had a tour of the major sites. Then the tour guide parked right outside my Budapest hotel and told the others they had a few hours to eat lunch/shop/etc. I grabbed my luggage out of the van and checked into my hotel. I stayed in Budapest while the others returned later in the afternoon to Vienna. It was a great choice for me... cost about $100 for the tour and the transport! Taking a cab to the train in Vienna, a train to Budapest, and a cab to my Budapest hotel would have cost about the same. So, I basically got the tour for free.

Enjoy your trip. All are great places!

My DH and I travelled to Vienna and Prague with our two teenagers a few years ago, and we have been back to Vienna several times since. If you are willing to spend a bit, we found the Hotel Konig Von Ungarn in Vienna to be absolutely charming. The lobby was beautiful and unique, and comfortable, lovely to sit in and relax. We had a "family" room, which consisted of a nice double room and bathroom, with private stairs leading up to another room for the kids, two twin beds with curtains enclosing each bed separately, very cool, and their own bathroom as well. We all remember the fabulous breakfasts at the hotel, which is very centrally located around the corner from the Stephansdom.

We took the train to Prague, stayed in two rooms at the Grand Hotel Praha, which was nice. It was located very centrally, maybe too much so, in the main square, directly across for the famous clock. It was a very busy and fun, but sort of noisy area. To address your question, we went on a half day walking tour of Prague's famous Jewish Museum, which was very interesting and enjoyable for all of us. We used Wittmann Tours, which I found after doing lots of Internet research. Our guide was great. The web address is www.wittmann-tours.com. As an aside, I remember they also offered a tour to Thereseinstadt,which I still regret we didn't have time to take.

Our son now lives in Vienna, so if you have more questions I would be happy to try to help, especially with restaurant recommendations and off-the-beaten path things to do.

Try to leave one full day for a day trip from Vienna to the nearby Wachau Valley, one of the prettiest stretches of the Danube anywhere - carpeted with vineyards and cute little towns, castles, etc - boats are a popular way to do it - about a two-hour cruise between Melk (famous abbey) and Durnstein/Krems. Easily done on your own.