As above really. Whitby/Scarborough are probably the most popular. Scarborough is like the NE Blackpool, Whitby is smaller and has more history.
Not sure how old the children are but Staithes is the location of the BBC programme 'Old Jack's Boat'
Salturn is nice but theres not much to do, Victorian pier and cliff lifts. The Chippy and beach is nice though.
If you venture to the South Gare (not something I'd visit) you will have a nice view of the mouth of the Tees, old steel works, a nuclear power station and the windfarm at Redcar. If you do a lemon top from pacittos on Redcar sea front is a must.
The beach between Redcar and Saltburn is nice at the Stray (Marske). There's a cafe and free parking along that stretch.
Not sure how far south you want to travel but Flambrough Rocks are lovely and you'll see Puffins.
Theres loads of nice places to visit on the moors loads of nice little villiages. Goathland is where the TV programme Hartbeat was set. Dont forget the NYM railway which runs steam trains between pickering and Whitby.
Theres a nice old pu at Blakey Ridge called the Lion Inn, not to far from Danby which is popular with walkers doing the Cleveland Way.
If you fancy a bit of exercise there are some nice walks at Roseburry Topping/Captain Cooks Monument