Stonehaven Centre.
Stonehaven's heart is around the Market Square and the harbour adjacent to the old town. Be sure to track down (and Lunch at) the fully restored and truly beautiful 1930's Carron Tearoom, adjacent to the Carron burn. Visit the Toll Booth museum at the harbour, and wander around the Old Net Drying green in Cowie Village on the north side of the bay. If you find your way around these three sights you will have by default, taken the most picturesque route around the town centre.

Dunnottar Woods
These are at the south exit of the town, but a short 10 minute walk from the town centre.
The full spread of blue bells when blossoming are magnificent. Here you will find the open air stone "BATH" and the Shell house, Gallows Hill (still with the "Drop hole" visible and coved with an iron grill).
In inclement weather wear stout shoes, as it can be wet and slippery underfoot.