Hi there! I live in Harrogate, not much going in the way of freebies but here are some ideas:
Valley Gardens - nice walks, crazy golf, paddling pool (depending on when you come), feed the ducks and it's free
Fewston resevoir - in the Yorkshirte Dales, loads of fresh air, fab views and walks, feed ducks and Canadian Geese
There's a few farms near by that are worth a visit, Ripley is a nice village for a day out and there's a big castle and a lovely ice cream shop
Knaresbourough is also a nice market down, boat trips along the river and Mother Shipton's Cave is worth a visit.
National Railway museum is a good day out and free as someone has mentioned as is The Royal Armouries in Leeds.
Harewood house is good for a dry day as they have a huge play fort but it is quite pricey - but they have a bird garden and penguins too. Fountains Abbey is another good choice but again it's National Trust so not cheap.
HTH.