Stonehaven is a charming seaside town, with some really good fish restaurants, but other than visiting Dunnottar Castle, there's not a whole lot to do. There are two charity shops and one very good music shop, plus a good beach walk. The walk out to Dunnottar is spectacular, especially if you go by way of the cliffs and the war memorial.
If you're traveling by car, the Fowlsheugh RSPB Nature Reserve, a few miles south of Stonehaven, is definitely worth a visit. There are thousands of seabirds of all kinds nesting on the cliffs, and there's a good chance of seeing puffins.
If you're taking the train, I'd recommend only one night in Stonehaven, and maybe one night in Arbroath. If you'd like to spend two nights, you could plan to arrive mid afternoon, explore the town, and hike out to Dunnottar the next day. Then you can catch an early train back to Edinburgh after your second night in Stonehaven.
If anyone in your party is a golfer, the Stonehaven Golf Club is one of the most spectacularly located courses in Scotland. It's a short walk north of town, just past the outdoor swimming pool. Even if no one in your party is a golfer, the location is absolutely stunning, and you can enjoy watching golfers on the first tee hitting a slice in to the North Sea. If anyone does want to play a round at the course, the green fee is 40 to 50 pounds, depending on the day of the week, and club rental is 10 pounds. In addition, the Cowie Chapel (ruin), just below the golf course, has many early gravestones. The clock tower, near the harbor, dates from about 1790 (post Outlander).
Arbroath is larger than Stonehaven, with an attractive harbor area, lots of restaurants close by the harbor, and a huge abbey dating from 1178. The Declaration of Arbroath, signed there in 1320, was the inspiration for the American Declaration of Independence.
p.s.: If you are taking the train, Stonehaven /Dunnottar could easily be done in a day, as there is a train once an hour out of Edinburgh. You could take an early train out of Edinburgh, and then a late train out of Stonehaven. The southbound trains run up 'til about 10:00 p.m.