Anywhere you go outside St. Petersburg will give you new perspective on the city's geography and its place in Russia's history. Whether you want to breathe fresh air, delight in country palaces of the nobility, or see rural architecture, branching out from the city is well worth a day or so of your trip.
Peterhof, Pavlovsk, and Pushkin (Tsarskoye Tselo -- home of the ill-fated Amber Room) house three of the most spectacular summer estates of the Romanov czars and most tour groups and individual tourists visit at least one of them. I recommend the same, though avoid them on rainy days. The extensive, landscaped parklands around the palaces are as much a part of the trip as the overly gilded interiors. The towns, charming in their own right, nevertheless remind you why Peter chose to base his capital elsewhere, on the river-crossed delta opening to the Baltic Sea.
Vyborg is a 13th-century town that has been Swedish, Finnish, and Karelian (an ethnic group based in what is now northwest Russia and eastern Finland) as well as Russian, and its architecture and mind-set are nothing like those of Peter the Great's Enlightenment-era capital. Kronshtadt is an island fortress that once served as a Viking rest stop but later became a major Soviet naval base closed to outsiders. It's now open to tourists and makes for a fascinating day trip.
All these destinations are best appreciated with a tour guide unless you know a lot about them before you go. Peterhof and Kronshtadt are fairly easy to reach on your own, and both can be accessed by boat in the warmer months. The other destinations require multiple modes of transportation, making them better candidates for an organized tour.