Strandvägen is probably the city's most elegant boulevard. It leads along the southern bank of Östermalm and overlooks part of the harbour (Nybroviken). The land side is occupied by belle époque buildings, many of which contain upscale hotels and restaurants and posh shops. The street is good for shopping if you have money to blow - there are for example shops with top Swedish design articles. If you don't have money to blow, visit the shops like you'd visit a museum, just to look at these many pretty things. For everyone the street is fine for window-shopping and people-watching.
Saluhallen "the market house" in the neighbourhood of Ostermalm (Östermalm) is so pretty. Here you can shop for delicacies or just have a stroll.
Secret tip: There is a free public bathroom, really clean also, where you can sneak in. (But don't tell anyone!)
Directions: The metro station is called "?stermalms Torg". Just get off the train there and ask your way, it's really close...