I am a recent grad of UC Davis, who lived and went to school in Berkeley. I have lived in Rockridge area, North Berkeley and went to highschool downtown
Here's the Wikipedia list of neighborhoods: en.wikipedia.org/wiki/List_of_Berkeley_neighborhoods
I would want to live somewhere near one of the great little streets that has shopping and restaurants:
Solano Ave. in North Berkeley—more expensive area, but there are little niches where you can find an apartment or a house for rent.
Hopkins—near Monterey Market (on their website they highlight the neigboring businesses as well: www.montereymarket.com/neighbours_HopkinsStreetBakery.html), there is a little strip of shops here that are really cute.
4th Street- is somewhat of a ritzy area but has the artist vibe going on- there is a new Crate and Barrel "C2B" that is Ikea meets Design Within Reach, a Petes coffee, and lots of high-end shops mixed with normal stuff--
Elmwood- this is getting over towards the Oakland area a bit- but Elmwood is awesome, and upcoming! They have very cute boutiques and have things for all ages.
Telegraph Ave near Berkeley Campus and Bancroft-- if you head up on Bancroft above Telegraph there are some apartments and such that are pretty cool, and you then can just walk down for shopping and food on Telegraph-- or down a few more blocks to Shattuck Ave (with all the movie theators and the bars as well as more restaurants, shopping and such). Will be in the student zone here with Berkeley High down the street and UCB up the street- and will encounter strange people :)
If willing to go into Oakland a bit—you must check out the Rockridge area. Great place to raise a family if North Berkeley is too expensive. Also lots going on. There is a rockridge bart station too, which makes for a very easy commute.
Also there is Emeryville- with some new apartments catering to the younger and or business crowd—but you would want to drive into Berkeley or SF for fun