Not currently in Philly but used to live in the suburbs (we have moved several times)
With a child who is not yet school age you have a lot of options. Chestnut Hill/Mt. Airy/Germantown is a bit further from Center City/University City, bigger houses, parks, quieter but still in the city. Keep in mind that as kids hit school age families in Philadelphia face difficult choices so you may find that friends move out.
As University City has gotten expensive, gentrification has spread west, but you need to have someone to guide you in exact locations. I have friends with kids in that area and it would be exceptionally convenient.
It also depends on what sort of housing you are looking for. Philadelphia is a rowhouse (terrace house) city for the most part, but there are also twins (semidetached) and of course apartments. The size of a typical row house varies quite a bit depending on neighbourhood--there are sections with blocks and blocks of identical houses. So if you are looking, say, for a larger house (I've been in a couple where I could spread my arms and almost touch both walls) they'll be in short supply in some locations. A lot of the houses in South Philadelphia (near the Italian Market and to the south) are very small but that area has tons of restaurants and stores and some people love it.
I would recommend a car but it is possible to do without if you live in the city. SEPTA (local transit) is decent, it's just hauling a toddler around that is tiring. If I didn't have a car I would want to live on a train/trolley line, though, and not rely entirely on buses. I can't think of any neighborhoods I think are really family unfriendly--there are ones I wouldn't recommend, but even Center City has families.