In the city limits Northwester and Chamonix do lessons. I believe both do “hunter jumper” lessons and Chamonix also does polo. Likely neither will come close to satisfying you if you were at all competitive/skilled. I believe if you part board a horse at Monastery someone can come in and teach. I don’t know any details. All 3 of these places, in terms of lessons/facilities are very basic type programs, from what I understand. Someone who has shown and/or really wants to progress in a substantive way will likely not be happy. IIRC these are all 3 very basic plop around, trail ride, maybe super super low level boots’n’chaps type show clientele. For someone who ever actually rode jumpers, I don’t suspect these will work.
There are better places in Chester County, Bucks County, Delaware, and South Jersey but they are not realistically 30 mins from Center City. Nothing is, unless you’re planning to go out at 2am. Anything quality in any direction will be 45 mins to an hour MINIMUM and maybe more with traffic depending on when you go. Parts of South Jersey are likely the quickest commute, though Ashford isn’t far. When I was a kid I recall Ashford being a good program but I have no connections to it as an adult. Woodedge, over the bridge in Moorestown, does lessons and has a pretty nice cadre of adult jumper riders and might be a good fit. Laurie J. is somewhere and her mom has a place in DE that you might look into.
You might find out where Penn’s eq team rides. I used to know but now I forget and it might be another good barn to try.
How tough are you? University City doesn’t bother me but I’m tough. Parts can be pretty shady but parts are also full of students and fine. Just make sure you’re in one of the good areas.
You can also live virtually anywhere on the el or Broad street line and take SEPTA to University City easily/cheaply/in a flash so unless you have a real desire to walk to campus there’s no reason to limit yourself to University City.