Here are some ideas:
If communist statues are your thing then head to Memento Park for a fascinating look into life in this former communist state. The park, born just after the 1989 fall of communism, was a dumping ground for residents of Budapest that sought to remove all the statues erected around the city. You’ll find Marx, Lenin and Engels, as well as Stalin’s boots, a tribute to the Stalin monument that was torn down in Budapest during the October Revolution.
Easiest way to get there: Take the metro 4 line to Kelenfold then get bus 101 or 150 to "Budateteny vasutallomas" (Campona).
Nestled on the right bank of the Danube river and sitting right at the border between Hungary and Slovakia, Esztergom is a beautiful little town with a rebuilt central square and home to Central Europe’s biggest church. It’s free to visit inside the basilica and you can pay extra to see the crypts, cupola, and the treasury.
You can also take a 500m walk across the Mária Valéria Bridge and cross into the Slovakian town of Štúrovo. You officially cross the border in the centre of the bridge, but because both countries are part of the Schengen zone there are no checks.
Easiest to get there: Get a MAV train from Nyugati station or take a Volanbusz bus route 800, 801 or 880-889.
A lovely little town at the foot of the Pills Hills, Szentendre is perfect for a blissful afternoon. For a small place it boasts 7 churches, all worth exploring. And just a little outside of town is the Hungarian Open Air Museum (Skanzen), a collection of old buildings from all over the country and a great way to get some insight into Hungarian life in a bygone era.
Being just 25km up the river from Budapest it also makes for a perfect biking trip. The route is easy to follow and gives you a chance to take in the suburbs of Budapest.
Easiest way to get there: Take the HÉV (suburban train) from Batthyány Square. Train leaves roughly every 10 - 20 minutes and gets you there in 40 minutes.
Located 30km northeast of Budapest, Gödöllő town centred around a royal palace and it’s charming gardens which are great for an afternoon stroll. Outside the palace there are cafes, shops and a market selling local produce.
Easiest way to get there: Take the HÉV from Örs Vezér tere. There are between 2 and 4 trains per hour and the trip takes about 51 minutes.
If you fancy tasting the sweet wines for which Hungary has become famous, then a trip to Tokaj is a must. It’s a town in the northeast of Hungary with dozens of wine cellar bars serving a variety of locally produced specialities.
You can literally taste your way around town until you’re feeling a little tipsy then head into a restaurant to try some Halászlé, a fish soup that’s a firm favourite amongst locals.
Easiest way to get there: Take a MAV train from Keleti station to Nyíregyháza and get off at Tokaj.