Sure Start do a lot of activities for under 5's in the school holidays, playgroups aren't usually on but they have activity days and day trips. Definitely contact your local centre for details of what's on.
There are a couple of good parks in Heaton, plus you have Leazes Park in the city centre. There's Adventurelands in Heaton but it's like hell on Earth when it's busy (it's a big indoor playframe).
The beach is a relatively short journey away, just hop the Metro to one of the coastal stops (Whitley Bay, Cullercoats, Tynemouth). Cullercoats has just a beach but it's small and has a lifeguard station so ideal for little ones. Tynemouth has a big park, a duck pond, and an aquarium as well as the beach, it also has lifeguards but is a much bigger beach. Whitley Bay has loads of shops near the beach, some arcades, etc.
For rainy days Newcastle City Centre has the Discovery Museum and the Great North Hancock Museum, they're both free and have lots to interest little ones (Hancock is nature and history, Discovery is science and local heritage), the Laing Art Gallery is also free and has a children's area. There's the Life Science Centre but it's a pay to get in affair and can be pricey (£10 an adult I think and about £6 for a child) however it's really good and has lots of interactive exhibits. Over the river is Seven Stories, a musum based around books/reading and they have lots of activites for children.
I can't think of any more off the top of my head but check with SureStart, check the local tourism board website, and ask about :)