Welcome to the Valley! Hope you feel settled soon!
Here is one of my favorite places: The Glendale Library. http://www.glendaleaz.com/Library/index.cfm
If you go to the site and look at the left hand side there is a link to "what's happening" and a sub-category of youth programs at the various branches. Most of the things are free and they are a lot of fun. Even if there is nothing scheduled, there are baskets of books and nice games and puzzles, especially at the Foothills Branch. Right next to the main branch on Brown and 59th Ave is also Saguaro Park with a huge play ground, grilling area and (my daughter's favourite) the historic ranch with free ranging chickens that like to get fed.
Then there is the Phoenix Zoo. We got an annual pass, so we can go on a whim and not worry about entry fees. That way we can also leave when either our daughter looses interest or it gets too hot or crowded. They open at 7:00 AM - perfect for the summer: Get there by 8 AM and leave by 10 : ) They always have something going on for kids. During the summer, they have a sprinkler play area which is great even for the little ones (as long as there are not too many bigger kids who simply do not pay attention...) http://www.phoenixzoo.org/
For a gazillion other things, check out the Parks and Rec site at : http://www.glendaleaz.com/parksandrecreation/index.cfm
And then there is an event site called Showup.com. They have a tab for kids activites and for free things, among many others!