Train - From Aberdeen it is the first stop South, and costs less than 5 pounds round trip.
Bus - 20-30 minute trip from Aberdeen even less expensive than the train.
Stonehaven is a very small town, so walking is your best bet
There are taxis in the area
Dunnottar Castle, . By far one of the most amazing sites in Scotland. The castle is perched on cliffs overhanging the North Sea and provides stunning views. The best way to get there for adventurous folk is the mile-long hike from the town of Stonehaven. The hake is a narrow path along the cliffs that provides stunning views the entire way. Definitely a must-see.
Stonehaven Harbor - The harbor forms what I would call the town center. There is a collection of local pubs and restaurants, though as it is a small town, we are talking about maybe 4 or 5 places. However, having a pint at a table overlooking the harbor and hanging with locals is a great experience. This town is very small, and in the town there are very few travelers, so it is a great chance to meet and talk to locals.
Play golf at the Stonehaven Golf Course.  Probably one of the most tremendous golf courses you will play in your lifetime. Absolutely stunning.
There is not much as far as things to buy here. I didn't see any gift shops, though I'm sure some souvenirs can be found.
There are a couple of gift shops actually - Purdies, in the Market Square sells an eclectic mix of gifts, cards, jewellery and shoes
Charles Michie, also in the Market Square is a pharmacy, but sells trinkets, gifts, ornanments etc
The Shell Hoosie at the harbour sells sea/marine based gifts as well as ice creams, coffee to take away etc
Can't go wrong at the local pubs..
There are some lovely cafe's as well - Mimi's, in the Market Square does the best hot chocolate around!
Here and Now, in Barclay Street, has lovely homebakes, sandiwches and snacks. Also sells fairtrade goods, gifts etc
Gino's, just up from the Market Square is a wee italian run cafe, and sells good quality food at reasonable prices
Molly Gunnings, at the beach is open all day and sells good Scottish breakfasts, as well as the usual homebakes, paninis etc
For World Class seafood try  The Tollbooth in the harbour.
In the town centre The Carron is well recommended 
If you simply want to buy produce then Charles McHardy in the Market Square is good , also Lembas Organic eggs are sold there .
Be sure to have a pint around the harbor and a summer evening.
Stonehaven should be more of a side excursion from Aberdeen than an overnight thing.