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.
Can't go wrong at the local pubs..
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.