Good to hear you're considering buying in Tucson. I just ran a quick search for the 85716 zip code with the criteria you gave and 18 results came up (I am including a link to the Tucson MLS) below so you can check for yourself.
As far as a strategy, I would say information is king. Find an experienced, helpful Realtor, one that you get along with, and learn as much as you can about the neighborhoods, comparable sales and the homebuying process.
As far as tips on neighborhoods, it really depends on what kind of feel you would like. For a home near amenities and within walking distance of many things consider living near El Con Mall. You could also look near Himmel Park which would put you near the edge of the University.
Hope this helps!