Help! Need a Preschool/daycare Facility for My Shy 3 Year Old Girl

The bottom line is:
you need to, personally go to a school, and tour it and talk to the Director about the school and their approach with kids.
You can't just choose one, without doing that.
And go with your child to see how she reacts to each one.
Yes it takes time.
But that is the way to do it.
I did that with both my kids, before I chose a preschool for them and enrolled them. And per my kids and their reactions to the school during the tour and how they responded, I chose a preschool for them. One that I was satisfied with while talking to the Director and personnel. So as such, my kids went to different preschools. Because, they liked different schools.

Both of my kiddos go to Bright Horizons in the ASU Research Park (right off Elliot and 101). I LOVE it! I have an infant there and an almost 3 year old so she'll be starting in their preschool in a month. She's been in daycare since she was 12 weeks old, but I just wasn't seeing the progress at her old center and with my new little one it was a perfect time to transition. I researched a number of centers in the area and just kept coming back to Bright Horizons (I visited 4 times and met all of the teachers they would interact with). They have a great staff and great teachers. Both of my kiddos are so happy there. My almost 3 year old runs to her teacher and friends on the playground each morning making drop offs a dream for me. When I pick her up, she can't wait to tell me about her day. They also have a wonderful curriculum. Each week we receive an email with what they'll be doing that week including books they'll be reading and projects they'll be doing. This has been great so we can track what she's learning and follow it at home as well. You also have a login to their site so you can view pictures that are taken in class, which I especially love so I'm not completely missing out. Another great thing is their menu. I was not a fan of the food she was eating at her last center. At Bright Horizons she's eating fresh fruits and veggies and just overall healthier meals. As you can tell, I'm extremely happy with our decision.

I only know of a few places in south Chandler. I would look at some private elementary schools (some offer per-k programs) or Montessori.

I don't know that area, but I will say that perhaps you should check out a montessori. That type of learning environment really helps shy kids to connect.