Pittsburgh is a lovely (and affordable) City, and you and your family are certain to enjoy it. The east end of Pittsburgh is full of nice neighborhoods within an easy (flat) commute from Oakland and the University of Pittsburgh. Squirrel Hill has the advantage of being just on the opposite side of Schenley Park from Oakland. The commute would be city streets, with some traffic moving at speed, but would be in or adjacent to a very large city park. The commute distance would be in the 3 mile range, probably, and very easy. As an added bonus, there are dozens of bus routes that connect the two neighborhoods, giving you a convenient option in times of bad weather, etc. The other really nice option would be the Regent Square area, which is located on the far side of Frick Park, a couple miles further east of the University. Same flat ride, same ride through a City Park for parts of the ride -- just a tad longer. If you want to look in the Regest Square area, my personal "would kill to live there area" would be the area between Braddock Avenue and Frick Park, south of Forbes Avenue.
I don't know where you are coming from, but you might want to know that Schenley Park is what I consider to be a traditional sort of very large urban park, with swimming pool, golf course, ice skatinf rink, etc. It also has some wooded areas with nice walking trails, etc. Frick Park is a more rustic park, sort of, with fewer built amenities aside from awesome CLAY tennis courts. Most of the rest of the park is wooded, with great trails for hiking, mountain biking, etc. I guess what I am trying to say is that I always find surprises in Frick Park -- a singletrack mountain bike trail leading to a hard packed hiking trail leading to a soccer field. On another end of the park is a big kids playground, or the clay tennis courts, or even manicured lawn bowling greens.
A little further north, you have the options of Shadyside, or north of that, Highland Park (yes, another large urban park, this one featuring the City Zoo.) Again, about 3-4 miles from the University, accessible for most of the way by relatively good cycling roads for Shadyside, a couple additional miles for Highland Park. Bus service to both areas is available from Oakland, but not quite as frequently as bus service to Squirrel Hill.