What to look for: FIT, FIT, FIT. How tall are you for starters? We can point you to what size range will probably fit you; smaller variables include whether you are long in the torso or long in the limbs, do you have wider shoulders, etc.
And then look for mechanics; does it brake, does it shift, does the rim rub on the brakes, do the tires/tubes hold air. If any of those are no, it's not a dealbreaker, likely it is quite simple to fix, but you have to either learn how to fix it (fun and rewarding for many people), or pay for a shop to fix it. Bring your own pump, if the tires are flat and the seller can't even deal with that, likely the bike is perfect and neglected. But if the tires are punctured (won't hold air when pumping), then you can't do a test ride. New tubes/tires is an easy fix though, and you should be able to check shifting, braking by lifting a wheel off the ground. If all those things are ok, then either the seller is enough of a mechanic to take care of those things (good), or they paid a shop to tune the bike up for sale (good), or the bike is essentially brand new and hasn't had time/use to get out-of-tune (good).
After that, look for extras: does it come with rack, fenders, frame pump, seatbag, helmet, bell, lights, gloves, shoes, spares (tubes, tires, wheels, pedals) -- I've seen bikes for sale with all those accessories before, but not usually. If the bike doesn't come with those things, are they at least possible to add-on? I.e. are there places to bolt on a rack and/or fenders if you want them? (in phx you can pretty well get away with no fenders)
As for parts, a somewhat-recent thread around here discussed the 'sweet spot' in the Shimano lineup and the consensus was that it is either Tiagra or 105, i.e. pretty much everybody agreed that the extra price of Ultegra and Dura-Ace gets you less weight, but not more reliability (probably less reliability); and below Tiagra you are saving cost, but sacrificing reliability.
Post some links to some PHX craigslist ads that interest you, this crowd loves to look at bikes and give their opinions about them!