I grew up in Phoenix so hopefully the following is helpful. Tempe is part of the Phoenix Metro Area and is located pretty much in the center so I may use the terms "Phoenix" and "Tempe" pretty interchangeably.
Tempe is a great part of the city, in my opinion. The parts of it closest to ASU have a lot of young people and housing is affordable. There are a lot of apartment buildings in the area as well as nice, older homes. Tempe's core is a bit older than other parts of Phoenix so the houses tend to not be so cookie cutter and many of the neighborhoods have nice old-growth trees and such. If you have 2-3 other people you want to live with, renting a house is very common.
Another advantage to living in Tempe and being close to ASU is you'll be close to nightlife and restaurants. Mill Avenue is the main street through downtown Tempe and is where many college students and people in their mid-twenties like to go out. Not far from that is Scottsdale which has its own thriving nightlife scene, but tends to be an older, wealthier, sometimes douchier kind of crowd. If you like art and culture, Phoenix is home to the Phoenix Symphony and Herberger Theater and big name theater productions come to ASU's Gammage Auditorium all the time. There are also a number of smaller orchestras and theater companies around the city if you prefer.
Food-wise, if you like Mexican food you'll be in luck. Good Mexican food is really easy to find in the Phoenix as long as you stay away from the chain restaurants. Los Dos Molinos is one of the most famous Mexican places in town, well known for its very excellent and very spicy food. It's almost always busy and they don't take reservations. My favorite joint though is a little hole in the wall called Los Taquitos in Ahwatukee (about 10 miles from ASU). It's delicious and cheap, try the carnitas. Since Phoenix is a major city you can also find just about any kind of cuisine you may want, and Tempe in particular has a good variety. Off the top of my head I would recommend Oregano's for pizza, Little Szechuan for Chinese, Four Peaks for a microbrewery/pub restaurant, Cornish Pastie Company for...Cornish pasties, and the Chuckbox for burgers.
One of my favorite things about Phoenix is how it is surrounded by a lot of little mountain ranges so no matter where you live, you're never too far away from good hiking, mountain biking, and rock climbing areas. In the summer it is very common to drive out of town and go tubing down the Salt River (Bring friends, sunscreen, and a cooler full of cold drinks. Be sure to have a designated driver because the cops will usually check every car leaving the area during the height of the season.) If you have a motorcycle, even better. There are a ton of great little towns up north that are really easy to do a day ride to just to grab some lunch and out of the heat for a while and the scenery is great. If you want to go further out, there is no end to good campgrounds and great natural sights. Obviously there is the Grand Canyon and Painted Desert but you also have the Petrified Forest, Mogollon Rim, Kartchner Caverns (one of my favorites and a very easy day trip from Phoenix) and a ton of others if you just do some research.
Unfortunately, you will need a car. Phoenix does have a bus network but there is only one rail line that starts east of Tempe and goes into downtown Phoenix and it shuts down at midnight. Free parking is very plentiful in the city, usually the only time you will have to pay is if you try to park downtown in Phoenix or one of the suburbs or if you are going to a sports game. Traffic does get pretty slow during rush hour but it's nowhere to the extreme of some places like Los Angeles. Car insurance is mandatory, but cheaper than in California.
All in all, Phoenix may not have the best restaurant scene or the best theater scene or the best nightlife scene of any city in the US but you can find pretty much whatever you might be looking for there if you do your homework. I think it's got a nice balance of big city amenities without the extremely high cost of living and overcrowding of a place like Los Angeles or New York. Yes, the state government is pants-on-head crazy but living in Arizona isn't a post-apocalyptic free for all like the media sometimes likes to paint it as. Hopefully this wall o' text didn't scare you off and had some helpful information for you. If you've got any other questions for me, I'll do my best to answer them.