Five places in the outskirt of the city you must see.
Gödöllő is famous for its beautiful Palace and garden. it’s a bit of Schönbrunn in Hungary. The palace at Gödöllő was originally built for the aristocratic Grassalkovich family; later Franz Josef, Emperor of Austria and King of Hungary and his wife Elisabeth ("Sisi") had their summer residence here. Gödöllő is about 30 km (20 mi) northeast from the outskirts of Budapest. It can be easily reached from Budapest with the suburban railway (HÉV).
Szentendre is another little treasure chest. This small town is the home of art and galleries. It is known for its museums (most notably the Open-Air Ethnographic Museum), galleries, and artists. Szentendre has been the home of many generations of Hungarian artists since the early 20th century. There are many museums and contemporary galleries representing the rich traditions of the visual art. Szentendre is in the northern outskirt of Budapest, also can be reached with the suburban railway.
Délegyháza is famous for its small lakes: the former mining area has beautiful and crystal clear lakes. There are both free and paid, and also a naturist beach, which makes Délegyháza a true paradise in the summer. If you don’t have time to travel to Lake Balaton, than choose this opportunity, because it’s only 27 km form the city. And, since 2008 the village is the site of a Trail of the Whispering Giants sculpture, the first of Peter Wolf Toth's sculptures located in Europe.
Visegrád is a small castle town in Pest County, in the north of Budapest on the right bank of the Danube. Visegrád is famous for the remains of the Early Renaissance summer palace of King Matthias Corvinus of Hungary and the medieval citadel. From the upper castle one can admire the panoramic view of the Danube Bend. Visegrád organizes Palace games every ear, which is the biggest feast of mediaeval fans.
Pilis is a beautiful hill (and also a village) in the outskirt of the city with many small settlements. It’s an extremely popular vacation destination in Hungary because of its lookout towers, hiking trails, and natural wonders like Rám Canyon, Thirring Rock, Turul Roundroute, Ferenczy Rock, which is believed to be the Heart Chakra of Earth, Preacher’s Stool with its gorgeous view and Pilis Summit, the second highest point of Transdanubia.