i currently live in squirrel hill, and i definitely recommend this neighborhood to anyone considering buying a house. it is only a 10-15 minute drive to magee/presby/montefiore, and it has lower property taxes than shadyside. there are plenty of things to do in squirrel hill, and another plus is that it is quiet.
if you're just looking to rent an apartment, i would suggest shadyside, squirrel hill, or the southside. like someone else mentioned, nobody lives in the downtown area, because there is nothing to do there.
here's my scoop on the different neighborhoods:
shadyside: hip area with cute stores on walnut street. also lots of bars in this area and close to the hospital (10-15 min drive). the negatives are higher property taxes and overall more expensive housing than areas such as oakland and squirrel hill.
oakland: medstudents live in a place called ruskin hall. there are also condos on streets such as dithridge which are pretty decent. otherwise, i don't know too many people who live in oakland, other than undergrads. housing is pretty cheap here and depending on where you live, you can walk to magee or presby/montefiore. i would personally not live here though because of the undergrads.
squirrel hill: great neighborhood with parks (schenley park and frick park)
and lots of trees. generally a quiet neighborhood where many families live. forbes ave and murray ave in squirrel hill have many restuarants and stores. i think it is probably the best close neighborhood to the hospitals. you will need to drive to the hospital though because buses don't run very often in pittsburgh
southside: very hip, up-and-coming area with the new condos off of carson st as well as the new southside works. has great coffee shops and bars. however, it is a bit too industrial for me--few trees, lots of warehouse-looking buildings. very close the hospital as well (5-10 min drive)
bloomfield: the little italy of pgh, with great restaurants and bars all over liberty ave. very cheap rent, but also not my favorite neighborhood because of the sparcity of trees and recreation. but still a great neighborhood. close to upmc shadyside (5-10 min), 15-20 min from magee and presby/monty
south hills & north hills: great neighborhoods with great houses for affordable prices. many families live out here, but i don't know too many people who are willing to commute to these areas. the ride is sweet in the morning (20 min) but if traffic gets bad it can take 30-40 min to commute.
i would personally live in the other areas i described above.
pm with any questions!