The Danube Bend north of Budapest offers some of Hungary's most rewarding day trips, and most are within easy reach by train, bus, or boat. Szentendre is the closest, a pretty riverside town of cobbled streets, Serbian Orthodox churches, and ceramics shops about 20 kilometers up the HEV suburban railway. Further on, Visegrád crowns the river's most dramatic curve with a hilltop citadel and the ruins of a royal palace, while Esztergom sits at the bend's far end beneath the largest church in Hungary. You can combine two of these in a long day, or take your time with one and be back in Budapest for dinner.
For something further afield, Hollókő lies about 100 kilometers northeast in the Cserhát hills, a UNESCO-listed village of whitewashed houses below a 13th-century castle where folk traditions still survive, especially at its Easter festival. It is a longer outing, but worth it if you want to see something of rural Hungary beyond the capital.