>>2289640Pilis mountains and Buda hills.
Budapest actually have some nice hiking locations. Last year they were infested with normies when hiking became a fad during the quarantine, but I heard the routes are ok again, since the average yuppie can't walk more than 10 kms. Normafa is a good starting point, plenty of buses can take you there from Széll Kálmán square. Hűvösvölgy is also good and more accessible, since it's the last stop of a few trams and the cog railway.
If you're adventurous, then go to Dobogókő via bus and hike to the Danube Bend on the blue trail at Visegrád and take the bus back to Budapest (or the other way around, but in this case you'll have to go up more than down).
This map shows all the trails in the country:
https://turistautak.hu/maps/mapnik/geojaunt.php?lat=47.15237&lon=19.62158&zoom=8&layers=00B00&reset 