Or will I be spending too much time traveling? Should I just do one or two, if so which do you recommend? Thanks.
If you have a full 10 days for seeing the 3 cities it can be done. I lived in Budapest and traveling by train to Vienna is easy and only a short train ride, 3hrs at most. Prague and Budapest are very similar. I spent 2.5 days in Prague and I got a good feel for it in those days. I took a night train to Prague from Budapest. Of course you want to spend as much time as you can in Budapest, slightly biased! They're all great cities!
I say yes also, but my bias is a bit different. I spent 1 month in Budapest and think I'd spend a bit more time in Vienna if I had to do it again. Of Course Budapest is cheaper, so that helps if you're thinking budget. The travel time between those is 4 hours Budapest - Wein - Prague and about 8 hours from Prague to Budapest but there is a good night train leaving around 11 pm and getting in around 8 a.m. Perfect night!
We did it and it was perfect for an overview of each city. Be sure to allow time in Budapest for all the mineral baths, cheap massages, etc. at Gellert Baths.
You'll have a ball!
If the 10 days includes your day of arrival and departure, I think it would be a stretch. All three of the cities are great and have their own unique personality and charm. If you want to "stop and smell the roses" and truly get a flavor of Budapest and Vienna, skip Prague and see it on your next trip. Prague is also a great town. None of the cities should be seen in a rush.
Hungary is fascinating. You can visit cities outside of Budapest (Eger, with it's Valley of Beautiful Women) and cities up the Danube (Szentendre) can certainly fill 5-6 days. They're an inexpensive boat or train ride from Budapest. My personal preference is to avoid rushing a visit. I also feel that I might never return and don't want to miss anything if I can help it.
You should be ok if you have 10 full days, We visted Prague and Vienna but not Budapest. We are planning on going their possible on our next trip this fall. Prague was wonderfull and cheap! Vienna was just okay for us. Like many other large european cities. It was very expensive. If you are wanting to do shopping do it in Prague. Stay longer in Prague if possible. Good Luck