I know that they do story time at the Barnes and Noble here in Scottsdale and I think they have it at many of their stores. The times are listed on the website if you want to check with stores close to you. You should check with the libraries too as I know the Scottsdale libraries also have story times and puppet shows and the like.
There's an indoor playground at Paradise Valley Mall that is set up for toddlers. My son loves it there and we've been going since he was just a couple of months older than your son is now. He asks to go to "the stores" all the time. There's a little slide and stuff the kids can crawl on. It's all padded so I always feel that my son is safe. It's right next to JCPenney. They do close it down for an hour at 2 or 3 (I can't remember which) in the afternoon for cleaning, so I wouldn't go right at that time.
I don't know what your budget is like but you might also consider a membership to the zoo. It cost me $50 for a year membership for me and my son and we can go as many times as we want for the whole year (plus you get 2 free passes to take somebody else along). They have a water play area and a petting zoo, which are his favorites right now and when it cools down a bit I imagine we will spend more time checking out the other animals. They also have special events scheduled for kids. They have a list of events on their website. It think they usually last for an hour or two and cost about $20 (you get a discount with a membership). We haven't actually been to any yet but we are signed up for a couple later on in the year. If you do decide on the membership, they will send you a magazine periodically where they advertise events going on. For example, a couple of weeks ago they shipped in 24 tons of snow and just let the kids play in it. It was free with zoo admission. http://www.phoenixzoo.org/
I also hightly recommend the McCormick-Stillman Railroad Park. They have a great playground and a train that you can ride around the park and a carosel (both cost $1, but admission is free). My son just loved watching the train go by while we sat and ate lunch. http://www.therailroadpark.com/index1.html
I hope that helps! Have fun!
B.