There are many activities in Aberdeen for kids of all ages.  Try to pick up the book "Aberdeen for Little People" (published by Yackiti Yack Limited) - you can find this at Waterstone's Bookstore.  This book is filled with ideas for children's birthday parties, restaurant recommendations, day trips, etc.
 
Visit the castles around Aberdeen - and be sure to check the Upcoming Activities - at times you will find special tours geared toward children, and ghost stories around Halloween. The Beach Esplanade - http://www.nobleleisure.com/ has plenty of places - bowling, ice skating, foam factory - for kids to play, as well as restaurants, movie theatres that parents can enjoy.
 
The local gyms - Kippie Lodge and David Lloyd's (membership fees required) have a host of activities, classes (both private and group) and summer programs for children.  Check out their websites for further information.
http://www.kippielodge.co.uk/
http://www.davidlloydleisure.co.uk/

Younger children will enjoy a stroll around Storybook Glen in Maryculter:  www.storybookglenaberdeen.co.uk 

Easter Anguston Farm in Peterculter is a special needs community that welcomes visitors.  You can visit the farm animals and the education center year-round and in the summer, you can pick your own fresh fruit.   http://www.vsa.org.uk/anguston-visiting.html

Hazlehead Park has a nice children’s playground, lovely gardens, a café, and a petting zoo.  http://www.aberdeencity.gov.uk/Parks/ns_pos/pos_HazleheadPark.asp

Duthie Park has children’s play equipment, winter gardens (indoor greenhouses with lovely plants), a cafe, a reflection pool with kayaking in the summer, and has easy access to the bike path (old railway line that heads to Banchory).  http://www.aberdeencity.gov.uk/Parks/ns_pos/pos_DuthiePark.asp

Dry Ski Slope in Alford/Garthdee:  Children will enjoy taking ski/snowboard lessons on the dry ski slope or you can rent intertubes at designated times of the day:  http://www.aberdeenshire.gov.uk/sport/skicentres/alford.asp, Garthdee:  http://www.skiaberdeen.com/

Older kids might enjoy paintball out in the country near Drumoak at Skirmish:  http://www.aberdeenskirmish.co.uk/

Adventure activities for the whole family at Deeside Activity Park:  http://www.deesideactivitypark.com/