When Pete says to Myka "Old chum... (welcome to the Pete Cave)" he is doing an impression of Adam West's Batman from the 1966 series... using the phrase West made famous when often referring to Robin, the Boy Wonder.
As Pete, Myka, and Claudia enter the supermarket, Pete says, "Alfred, let's go shopping," That is what Bruce Wayne says to his butler after they discover The Joker's poisoned products on a TV advertisement.