Core
Border
Persephone
Eavedown Docks
Verbena
Kettle
Rim
Beylix - garbage dump
Newhouse - pick up sherif
Regina
Frisco - mining town, "cursed" plat coins
Angel Hair Comet
The Lullabelle Mine - ice mine
Torval Station - bigest port on the rim
Cowbow Feng's Space Bar and Grill
Madame Shi Shi's - brothel
Badwater Station - Refueling spot
Three Hills
Oak Ridge - small town - "Gonna be a Hanging"
Specific info on various worlds Firefly WIki