West Cliff has mini golf, paddling lake, go cart, boating lake, children's activities, and of course the famous Whitby Wizard- science experiments for all the family
Pavillion Spa complex often has indoor things going on- stories/films/workshops etc.
In Whitby- Airy Hill Play Park is a good one.
At Ruswarp there's an indoor play place, ( and outdoor stuff too) [http://www.minimonsterz.co.uk/ mini monsterz]]
[http://www.redcar-cleveland.gov.uk/main.nsf/Web+Full+List/FF0C456F8B6AD7E780256BBF003C8 FCA?OpenDocument The Moors Centre, Danby]] has a lovely outdoor play area, and tracks through the woods too]]
We go to Sandsend for beach- it's lovely. There's a stream which runs into the sea, and a small lake forms where it's shallow and safe to sail in a dinghy.
Children love climbing the steps to the Abbey.
Outside Whitby, the moors around Hole of Horcum are spectacular. There are walks and tracks, and children can charge around in the heather without hurting themselves.
Scarborough is good too, but I don't think you'd need to go there when there's so much to do in Whitby, ( and Whitby is much prettier)
There are some great places to eat,- The Magpie, ( but you'll have to queue), White Horse and Griffin- on Church Street never lets us down,- The Moon in Sixpence, Marine Parade.