USA to Prague or Budapest first? Which makes sense? and other questions

Fly so called open jaw to avoid backtracking. IMHO it does not matter which city you fly first. In Budapest Pest is flat and Buda is hilly. In Prague Old Town is flat and Mala Strana (Little Quarter) and Hradcany (Castle) are hilly. You can either skip the hilly parts or take public transportation or taxi.

When I visited those two cities about 6 years ago I was fit, but since then I have undergone surgery that left my right side weak and causes me to tire easily. I often think about how do-able travel might be given the opportunity. So I just wanted to add to Ilja's helpful descriptions of the cities how do-able I think they would be for me as an example. Old Town Prague is flat and compact which is good, but many of the streets are cobbled, which is charming, but hard to walk on. Plus, the streets are often veeeery crowded which is also tiring. I would suggest making good use of the little cafes. Walk a little, have a tea or coffee and people-watch a while, and go a little more. I think Budapest would be easier. It's more spread out, but far less crowded. They also seem to have an abundance of benches so it's easier to sit a bit and take a quick breather. And as Ilja said, the Pest side is flat, and unlike Prague doesn't have cobbles. Many of the places of interest on the Buda side are right near the funicular up Castle Hill. So, though that side is hilly, once you get up on Castle Hill, there is quite a bit to see without having to go up and down. However, I'm not sure how far the funicular is from the closest subway station. BTW, the subway system in both cities is easy to negotiate and goes most everywhere you want to go. However, I don't remember how common escalators and/or elevators are. I think they are both do-able if you take your time. In fact, taking things more slowly, not rushing from site to site, sitting in the little squares and cafes and taking in the ambiance, you will probably enjoy it more. As for order, perhaps Prague first while your mom is more rested, then Budapest to unwind a bit after the bustle of Prague. Have a great trip!

Budapest Metros - the two new lines have long escalators at each entrance, they are necessary as these were built as bomb shelters (from the US/NATO) and are really deep (check out the giant turbines circulating the air in the stairways). The old original Yellow Line does not, but it is barely below street level, as in walking down your stairs to your basement. Even without stairs for the Yellow Line, where it interchanges with the other lines the service there is with escalators.
The base of the funicular is not serviced by the metro, but is reached directly by a bus coming across the Chain Bridge. There is also excellent tram and bus service throughout Budapest. Do not forget to validate your ticket on entrance to the Budapest Metro. At the major stations we found they had personnel at the entrances to watch for this. In fact, we were no more than 20 seconds in motion on the bus from the airport on arrival when armbanded transit police on the bus approached us to check for our validated tickets; I imagine that checking obvious non-locals boarding incorrectly at the airport is an easy revenue source. FWIW, we never noticed any ticket checkers the few times we used the Prague system.

I would also advise you not to rely on using taxis in Budapest. The taxi drivers are notorious cheats. They will even cheat you on the meter and take 3 or 4 times the correct fare. When I wanted a taxi from my hotel, they would order it and give me an estimate of the fare and the drivers I got were honest. But I did get gouged once taking a taxi back to my hotel. I didn't have occasion to use taxis in Prague. You can get a lot of (free) information and valuable advice by using the forums on tripadvisor.com. That along with a good guide book or two should do it.

There is good train and bus service between the two cities. With 2 weeks, you could even stop for 2-3 nights in Vienna on the way. Trams are an easy way to get around all these cities.

Thank you ALL so much. It seems now the plot thickens and my mom is talking Vienna, Prague and Budapest and taking a river boat as well. All this in two weeks seems a lot to me but she is 82 so she gets to decide! I have been looking at Airbnb for places to stay. Any advice, info appreciated!
Thanks, Carol

You can take a boat (hydrofoil) from Budapest to Vienna as we did. Then dinner river cruise in Prague. It's magic.

The Prague ticket checkers are there. And its not true that Czechs get a free pass from them, as my Czech husband is routinely stopped and I, the American, have not been stopped in the Metro transfer points over the last 11 years. Maybe its because I'm usually the one that looks like the Queen Mary steaming through on a shopping mission or to entertainment on time & can't be bothered. My husband tends to trail behind me and makes a target of himself. The only time my ticket was checked was on a tram when every single passenger was asked for proof no matter who they were. You can still do the hilly sections of Budapest and Prague as long as you take it slow and take some transportion from point A to B. Use the trams, busses, and taxis, and stay off the Metro. Even though there are quite a few escaluators, there still are a bunch of stairs in many of the stations which is really difficult for anyone with mobility issues. Vienna is a really easy town and I recommend staying at Pension Suzanne, which is a very short walk to opera and shopping zone. My suggestion is to do some research using sites like Rick Steve's and Trip Advisor to narrow down hotels in your budget that have an elevator which are close to major sites, google your picks, and then contact them via their website or email address for the best rates.

Including Vienna is a great idea. Much more to do and see in Vienna than Budapest. Prague is awesome. If you want to fly, get an air ticket into Prague and out of Budapest. Use trains between the cities. Also in Prague, make sure your hotel is on the Old Town side. Flat and easy to walk around in. The castle side is hilly. Same goes with Budapest.

Have you considered any of the Rick Steves city tours? There's one for Prague-Budapest and one Prague-Vienna-Berlin. May be too much walking for your mom though. I would go to Prague first because the later you get into June, the more crowded it will be. Also, might consider a day trip to Cesky Krumlov from Prague. Beautifully preserved medieval/Baroque town. Probably not as crowded as Prague. Can't recall offhand how hilly it is though. Hope your trip is fantastic. Ross in Denver