I'm living in a condo downtown in the arts district/roosevelt row area (4th ave/McKinley) and love it - I can walk to everything and it's a very low traffic area. There's two parts to downtown really - the business part and the historic district/art community part.
All the bars/restaurants I go to are cool little bungalows that are turned into bars and what not. First Friday is a two minute walk.
Cibo, Local Breeze, Royal coffee, lola coffee, fair trade coffee, pasta bar, sens, moira, roosevelt, lost leaf, conspire, breadfruit are all a few blocks away. Suns stadium/Theatre/Diamondbacks/Phx library/Art museum/Heard museum/Science museum are about a mile away, again an easy walk, but even quicker with the lightrail nearby which will also take you down into ASU area.
What do you like to do for fun? If you want to go cougar hunting than Scottsdale is your friend. There's a lot of good food too, but same goes for downtown.
I lived in tempe for a year and liked it, but there's simply too many lights/driving to get anywhere. Same kinda goes for Scottsdale, you'll be doing a fair bit of driving. On the upside if you are really into hiking Pinnacle Peak (which is my favorite climb) is right there.
I pay 850 for everything included for a pretty spacious 1br/1bath w/ nice appliances and whatnot. I live by myself and living downtown can be a bit of a stretch on a budget, but if you get a roommate your options open up quite a bit.
Craigslist will be your friend in this case for certain areas of Phx since so many apts/houses are individually owned.
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If you are "young and single" there is a downtown asu campus and unless you're hitting up oldtown scottsdale you're gonna run into the more douchebag/club crowd/older people/cougars in Scottsdale...I mean think about it, Arizona as it is is a retirement state and Scottsdale is one of the ritzier/nicer areas to live.
Tempe will have your youngest crowd if that's what you're really interested in though.