I would stay in Olde City and check out the Constitution Center as well as Independence Hall. Both are interesting spots and, honesly, you can spend quite some time at the new Constitution Center. Both are within walking distance to the Betsy Ross House (and the former Real World house) at 3rd and Arch as well as Elferth's Alley. Also, you could walk over to Market and in between 3rd and 4th Streets for Franklin Court. It's Ben Franklin's old residence as well as an underground (kind of cheesy, but somewhat fun) museum.
If you want a funky vibe...head to South Street and Penn's Landing area for some interesting shops, decent coffee cafes and some bars. You can walk over 95 on a pedestrian bridge and it gives you a nice view of the Ben Franklin Bridge and Delaware River.
I don't think I'd reccomend going over to Camden...but the PATCO (located at 8th and Market) is only one stop away from the University area of Camden. That area is not really dangerous and is heavily polices by three college campus police forces (actually no joke) and Camden police. The Aquarium and the USS New Jersey are a few (relatively safe and touristy) blocks away.
The Zoo is actually nice...if you're out that way...you can head over to Kelly Drive and Boathouse Row. It's nature-y and a nice view of the Schukyll.
City Hall and the Art Museum are also walkable (though a little long) to each other. City Hall is located roughly at 15th and Market/JFK. It's good for a photo op, but not too much else. The Art Museum is actually quite nice...but try not to be one of those Rocky people who run up the steps and jump in the air.
Let me know if you need any other suggestions...such as shopping, nightlife, other historical sights, etc.
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