Traditional Gyoza shops or shops in front of the station can get very crowded during peak hours. So my advice is to arrive before noon (aim for around 11am).
Taste the first piece plain before reaching for the sauce (tare).
Track down one of the city's creative takes, Jyugem's Shichifuku Gyoza or Satsuki's yuzu gyoza are good places to start, but places like Gyoza 365, Baris, Bariron, Gyotendo, Tonkikki are a good reminder of how far shops here push the format.
Gyoza in Utsunomiya is not limited to just meals. There are also many Gyoza themed snacks sold at souvenir shops in JR Utsunomiya Station.