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Orion Square plays host to various public events and festivals every weekend. Great place to stumble upon events if you have no prior knowledge.
宇都宮美術館
うつのみや妖精ミュージアム
An fairy-theme art museum.
飛山城史跡公園
二荒山神社, "Two Storm Mountain"
This is probably the most popular shrine in Utsunomiya. The front is lined by a massive stairway several stories in height before ending at a large temple. Excellent for pictures and very convenient.
八幡山公園
A popular spot for cherry blossom viewing (hanami). The park is quiet, relaxed, and usually pretty empty except for certain summer celebrations. It contains a tower-viewing area, a large foot-bridge, and a play areas for families. It is a worth a picnic if you have extra time. Within the park grounds is Utsunomiya's velodrome. For ¥100 entry you can watch keirin bicycle races on race days, or bet on the outcomes in one of the few institutions in Japan allowing legal gambling. (The gaming aspect of this is not recommended except for those who are fluent in written Japanese; there is no assistance for other languages.)
宇都宮城 Utsunomiya-jō
The castle is Utsunomiya's claim to Japanese Royalty, where the emperor or other royal members would rest and/or change horses before their last leg to Nikko. The castle is reconstructed from previous excavations and is an excellent place to picnic, but little else. Historical significance and exhibits are limited.