Val, when in SPb, check restaurant Palkin for a great Russian authentic food. Prices are high, but well worth visit.
For contemporary cuisine, simple but sophisticated enough, I recommend restaurant "Teplo" on Bolshaya Morskaya. The owner and all staff are very nice and friendly. Prices are affordable.
For coffee, tea, exquisite desserts, or light breakfast, and unforgettable interior and atmosphere of opulent grocery store (think Harrods food department, or Fortnum & Mason) in the beginning of 20th century, visit coffe shop at Eliseevsky gastronom on Nevsky, 56. It won't cost you a fortune.
For the very best Georgian food you must go to Moscow and book a table at "Darbazi" restaurant. Their food is simply to die for.