I will probably get in trouble for this, but I'm not going to answer the "which 2 -3" part of the question because I'm not sure what you would enjoy and I don't know your budget or means of transportation. Some are great by train, some do better with a car. Some are exhausting and some are laid back.
So here is my list and few observations:
Pécs is a very enjoyable place for an early morning train down and a late train the next day or early morning the following day. Its pretty unique. Its one of those places we run to when the September weather in Budapest turns cold early cause its generally a few degrees warmer in Pecs.
Eger is a great overnight trip. Longer if you are into wineries and wine cellars. You can pair it with Holloko (not too impressive in my book).
Gödöllő really isn’t an out of town trip. It’s a suburb trip. Best done in the spring when the flowers are blooming in the gardens.
You can do all three towns on a Danube Bend tour but it’s too fast for me and a bit tiring.
Szentendre is a good half day trip that I enjoy. Do Buda in the morning then catch the commuter to Szentendre in time for lunch there then catch the boat back to Budapest (about 5pm – look it up). Budapest is beautiful to enter on the water.
Visegrad is “okay” and much better if an event is going on.
Esztergom is one of the great overlooked destinations by tourists. It’s a very long day trip or an overnight event. I’ve only been once and I need to return for at least an overnight.
Vác is also a nice little typical village with some interesting history and mummies.. Stop and feed the bears on the way back (day trip)
Panonhalma and Győr well worth the effort (long day trip or an overnight stay either on its own or when traveling between Vienna and Budapest)
Tihany; considered by many the enjoyable and prettiest part of Balaton. I love it. Its an overnight or a two night trip depending on if you want to visit some of the other Balaton towns (all are good one way or another).
Komárom; one of the more fascinating place I have visited. But you have to be into old forts to enjoy it.
Etyeki, especially if you can be there for one of their Piknik’s A lot of fun.
A couple I haven’t been to yet, but on my menu are
Kecskemét (day trip to an interesting town)
Lillafüred, beautiful old hotel surrounded by a late and forests, forest narrow gauge railroad, caverns, nature.
Lázár Lovaspark I have never been to. Waiting until I want to deal with the tourists.
Okay, I highlighted a few of the more favorite stops.