Scottsdale AZ teacher here. Go to Mesa Public Schools and look at the boundary maps to see where the schools are.
Go to
http://www.azed.gov/research-evaluation/aims-assessment-results/ and you can find test score results which can guide you. Anything above a 90% is very good. 80% is decent. Below 70% stay away. But, they changed the math test in 2010, so you may notice scores dipped a bit that year. Once you get an idea of what schools are "good" you can go to greatschools.net for a little but more parent feedback, just keep in mind complainers are more likely to post that happy people!
You may also find schools labeled excelling, highly performing, performing, etc and A-F. They just changed the way they label schools to letter grades. You can be an excelling school and still get a B, if your kids didn't grow as much compared to the other excelling schools. This is a weird thing, so I wouldn't get hung up on an A vs B school as much as excelling vs performing. You want excelling or highly performing.
Mesa Public Schools is a HUGE district. I just counted and there are 56 elementary schools. Mesa itself is a really varied town, from very impoverished, mostly spanish speaking immigrants to a very concentrated Mormon population, to retirement villages, to a very wealthy area. It may be a daunting task to sort through unless you have an idea of where exactly you will be living.
Gilbert is very close by (depending on where in Mesa you are looking at) and has excellent schools (for the most part). It may also be worth looking into them.
In the East Valley (where Mesa and Gilbert both are) there is a large Charter School movement. I personally am against charter schools because they take public funds, yet don't have to take special education kids, English Language Learners or have certified teachers, but a lot of people like them. Because they are regulated a lot looser, some are really good, some are really bad.
Good luck in your search!