I've lived in East Mesa for 7 1/2 years and have found caring neighbors, good friends, great recreation & entertainment options. If you like statistics the City Data site is a good read. Its a good idea to compare it to your current city.
When evaluating a neighborhood you should investigate local conditions. Depending on your own particular needs and tastes, some of the following factors may be more important considerations than others:
quality of schools
proximity to schools, employment, hospitals, shops, public transportation, prisons, freeways, airports, beaches, parks, stadiums and cultural activities such as museums, concerts and theaters.
I'm a former WA resident who bought in Mesa and we love it. I can't comment on the crime stats etc as previous colleagues have stated, but they gave you the same website I'd have given you to check out, azcentral.com. There is a page that will show a zip code and will map out where crimes occured in the past period of time (you can pick I believe) and if you explore the Mesa zip codes, this should give you some of the info you want.
Another tip I always give to my buyers, but especially relocating buyers, is to go do a bit of walking around a neighborhood you are considering. Knock on a few doors and ask how people like living there- believe me you will be fascinated by the information they give you!
Best of luck and please check out my website. If I can help you further, just fill in one of the contact pages there and I'll be in touch.