The big question is of course: what do you like? What type of zoos do you like, are you a rarity hunter, do you like fish/birds/reptiles, are you after exhibit quality and how much time do you have?
Given how "compact" Central Europe is and every major city is relatively easily reached by train it might seem easy to just visit multiple countries. That is possible, but I would advise to stick to one country/area (with maybe 1-2 sidetrips) and really do more than the big names. You will likely be back in Europe one day to visit another corner on the continent and see the big names there.
Berlin is an obvious starting point and there would be enough to see in the city for weeks. Zoo wise there are options for day trips from Berlin (or small loops). If you like fish, then head to Stralsund, which has Germany's largest aquarium the Ozeaneum, focusing completely on local marine fish and is very interenting. You could add Zoo Rostock as a 2 day trip, which is an interesting zoo I really like with a strong focus on naturalistic exhibits, but with GDR history still very visible.
South from Berlin, Leipzig is an obvious choice and if you are into rarities, both Magdeburg and Schoenebeck would offer an option for a long day trip. I would also consider visiting Dresden, a city with a beautiful historic city on the river Elbe and a surprisingly good, but low key, city zoo. The zoo could be combined in one day with Tierpark Goerlitz, which is one of the best small zoos anywhere and has a unique focus on Central/Eastern Asian fauna.
In Southern Germany Tiergarten Nuernberg, Tierpark Hellabrunn in Munich and the Wilhelma in Stuttgart are the 3 main zoos. All are of a very high level and offer something unique. Wilhelma is a unique combination of zoo, botanical garden and 19th century and 1970s architecture. It is lovely landscaped (with some brutalistic concrete buildings to compensate for all loveliness) and has a huge collection. Nuernberg has some of the most innovative exhibitry on the continent and is a lovely zoo to walk around in. Tierpark Hellabrunn is a solid zoo, slightly ABC zoo that makes the most of the riverine forest it is set in.
If you are into birds a day trip to Augsburg + Bird park Olching is very nice from Munich. A day trip to the Alpenzoo in Innsbruck is extremely worthwhile too. Set in the Alps, this montane zoo has the best collection of European fauna from mammals to fish anywhere in the world.
When in Germany, I would also consider visiting one of the many concentration camps that are now musea. While not a pretty part of history, these places are of huge significance and always make for impressive visits. Oranienburg-Sachsenhausen is easily done as a day trip form Berlin, and Dachau can be done from Munich. Alternatively there is Buchenwald if you also want to see the historic city of Weimar between Berlin and Munich.
If you insist on going to Budapest, it makes no sense not to stop in Vienna in between. A wonderful city, with one of the best zoos on the continent and is the oldest zoo in the world (and really makes a celebration of 270 years of zoo history).
You are probably already aware of
www.zootierliste.de to check which species is held where. And for trip inspiration, I am going to shamelessly self-promote these threads:
Europe's 50 must see zoos
Europe's 100 must see exhibits
One question, if you had to pick Berlin Zoo or Tierpark, which would you go for and why? I've been to Berlin Zoo once before but it was a long time ago and I wasn't a big zoo nerd yet. So I'm thinking if I end up only able to do one, to do the Tierpark... but the Zoo was really awesome as well and I kinda want to see it from an adult, zoological perspective. Argh. If only I had unlimited time and money! Lol.
