Budapest vs Barcelona with a bad knee

There is more than enough to see and do in Barcelona in a week. Other than Montjuic, the terrain is level.

I didn't find either Budapest or Vienna to have a lot of steep inclines. Those that are steep, like the hills and castle district in Buda are reachable by tram to avoid the worst of the hills. The Pest side is mostly flat. Vienna, at least in the central city, is not hilly.

Budapest is so well connected with trams that you can see most of the sights without much walking. Other than the trams the M1 metro connects a few of sights to a couple of tram lines.

I just came back from a few days in Barcelona. There's plenty to do in Barcelona itself. There are day trips, but I wouldn't recommend them because of hilly terrain (Tarragona, Girona, Montserrat). You could go to Figueres for the Dali Theatre and Jewel Museums (loved them).

How are you with stairs? Most of the metro stations I used had escalators but there were a couple of times I had to use the stairs. . . one flight. Also in Budapest, the original metro line (#1) is one flight below ground, no escalators.

Other than as others have detailed, both cities are mostly level and there's plenty to enjoy in both. I remember Vienna as also being level. Both are great choices. Weather is unpredictable, but Barcelona is more likely to be hot in June. Of course, last time I was in Vienna was the first weekend in June and there was a heat wave, temps in low 30's and NO air conditioning in most places, including my hotel room.

cannot comment on Barcelona but can comment with some expertise about Budapest and Vienna as my 80+ dad and I just returned yesterday from these two cities.
Dad is very active but he find walking long distances difficult and so we try very hard to minimize it and to save any steps to that which is necessary to see what he wants. Budapest was ideal. The tram system was excellent and most places were less than a one or 2 block walk or less. We were able to take the funicular up to the castle and there is a city bus that goes thu from the funicular to the stop near the Church. That bus continued on across the bridge to where we could get the tram to our hotel. The 2 ( I think ) tram is made for sightseeing and we were able to connect to that easily from the 4 - which was near our hotel. Using the tram and the metro we were able to visit the Szechenyl baths ( our favorite thing DO IT) heroes Square, Mathias Church, the Great Market and Great Synagogue . We saw other things using a bus tour . Be aware that the pavement was uneven and cobble stones. We did see St Stephens- but that was a bit of a walk from the train. We really enjoyed just riding the 2 tram at different times- seeing the buildings and bridges
We had only one day in Vienna. Again the tram system worked very well for us. There is some walking involved - esp in the pedestrian zones- that made it a bit harder . Dad uses a cane with a seat and we take advantage of ' rest stops'. For example, we took a longer time than most in St Stephens Church. We used the seating opportunity to pray, contemplate and light a candle while Dad also could rest. We had bratwurst from the Weiner mobile closer to the Opera House while sitting on a bench. In our prior trips, we would not have stayed as long in St Stephens and would have skipped the bratwurst and went straight to the Museum.
Looking at the transportation maps for Budapest and Vienna in detail will help you a lot. I wish I had taken more time to do that.
James' post here are excellent. One thing I learned on this trip especially is how much walking is expected. It was very hard to get people to understand that 'yes, we will take the tram just one stop or the train one stop' as that is not the norm. That meant I had to really plan out our day and really study the transit map in advance . We bought multi day passes which enabled us to just get on and off at will. It was worth it- we had a great trip and going slowly opened us up to some interesting experiences and conversations with local people.
While you didn't ask- may I suggest you request airport assistance at each of your airports. It was invaluable. Dad was meet with a 'wheelchair' at the jetway on arrival , and then assisted to the next gate. He was assisted thru Passport control and then customs both on arriving and leaving. Our flights connected thru Amsterdam and the travel assistant told us it would have been a 45 minute walk from our arrival gate to the next departure gate and he was not exaggerating.
My dad is a great example in many ways and ageing gracefully is one of them. On our cruise ship, he was up and dancing but realized he would not be able to walk the long distances between gates and so requested and accepted help. We also request hotel rooms close to the elevator and with no steps.
Hope this helps.

Thank you all for the info, and Diane your response helps a lot. My bucket list trip is Budapest, Vienna and Prague. My older daughter and I were scheduled to go there last year but my injury kept me from going. She and her husband went and loved it. I'll be taking my two daughters on this trip and they have both travelled extensively, so we decided to go somewhere where none of us had been. So, Barcelona it is for this trip. Budapest will stay my bucket list.

Thanks again!

We did Prague as well. Not as easy.
Glad i could help and Im sure you and your daughters will love Barcelona. Take notes!