L.S.
I have a two year old and we love to go to Encanto Park. They are located on approximately 15th Avenue and Encanto Avenue - kind of located near the downtown area but just slightly west. Website is Enchantedisland.com. They have a picnic area. Lots of ducks, geese etc so save your old bread. They have a small amusement park with train, merry go round and a few other rides that my two year old loves. Admission is free - you only pay for the rides you want. We actually did her birthday there this past year.
Another fun place is Rawhide. They are located in the east valley off of I-10. Website is Rawhide.com. Again, admission is free and you only pay for what you want. It is a western themed park. It is like walking down the street in an old western town. They have shops, pony rides, train ride, camel rides and a petting zoo. They also have a band that plays and the kids can get on stage and play pretend instruments. Again, my daughter just loved it.
Also, most of the malls have indoor playlands. There is also the Stillman Railroad Park located in Scottsdale. I am not sure of the location or web address but a google search should help you.
I hope this helps and have fun.
L.