In Center City, the best public schools are Meredith, McCall, and Greenfield. (Check them out here: http://www.centercityschools.com) In University City, there is the Penn Alexander school which is a really great public school. There are also private schools and charter schools which are listed on the website.
If you are choosing one of the public schools, you have to consider the catchment areas. If you fall within a catchment area, you can attend that school. THere is a website where you can map out the catchment areas here: http://cml.upenn.edu/nbase/ (although it is a little tricky to generate the maps, I found it very helpful.) You have to be really careful, because the catchment areas are very specific and their borders often run down the center of the street.
Obviously, if you choose a private school or charter, you don't have to worry so much about this.
We love the following center city areas for safety and good schools: Queen's Village, Bella Vista, the Rittenhouse Area, FItler Square, Society Hill, and most parts of University City (which is not center city per se). Of course, there are others, but these are the ones we have researched. They also trend to be pretty expensive. I have no info on the outlying areas.
Hope this helps a little...
That is a hard question to answer without knowing a little bit more about you, what your looking for in a neighborhood and your price range. I am a Center City Realtor and I also live in the Bella Vista/Queen Village neighborhood which is a great neighborhood with a lot of young familys and a great school (Meredith) but a 3 bedroom home is going to run you $300,000+ depending on the exact location and the condition. But if that is more than what you can spend there are a lot of great neighborhoods in the city. Feel free to contact me anytime to discuss what you are looking for in a house and neighborhood at no obligation.
You should start by calling a mortgage professional to get a preapproval on how much home you can afford. Once you have that in hand, the task becomes rather easy. There are many great areas to send the boys to school, but you have to narrow the field first by knowing where you can afford to look. I am a center city Realtor. I grew up in the burbs and moved here, like you are planning to do. If you get into new construction, you can get a tax abatement and you will be shocked at the tax savings relative to the burbs. Contact me if I can help you.
I am a real estate agent, and would gladly help you with your search. Please call with any questions. Alex