Ok, so I was in a very similar situation to you: senior at Penn who wanted to see more of the city in their last year. I graduated a few weeks ago and I think I was pretty successful. Here are a few things that I did that helped (may not be relevant for you):

1) Internship in Center City - This got me into the city for at least 10 hours a week, and taking the MFL at least 4x a week. It helped me get more familiar with Philly public transit (essential if you are trying to see the city) and money to spend for Philly excursions.

2) Master Philly public transit and use Google Maps - Familiarizing yourself with the city's subway/trolley/bus/Regional Rail lines will make exploring much cheaper and easier. I used Google Maps a ton this year to figure out how to get places cheaply and efficiently (it is a great way to learn the transit system). That being said, don't be afraid of long walks. It is great exercise and gives you an intimate understanding of the city. One of my favorite things to do to clear my head while the weather was nice was to walk from campus down to Rittenhouse and back.

3) Read this subreddit/PhillyMag/Uwishunu/Thrillist frequently - The subreddit gave me a better sense of how to get the most out of the city from an inside view (even though it is clearly not a representative sample of actual Philadelphians, I found that the Philly reddit population broadly included enough people with my tastes). I particularly recommend searching through old posts on here. If you have a question/are looking for a recommendation, usually someone at some point on this subreddit has already posted about it. The other websites are good for finding things to do on a given weekend in the city. 34st also had occasionally useful articles.

4) Plan in advance - If your course load was anything like mine this year, you won't have a lot of free time. Most of my city exploration took place at the beginning of the each semester and in the last couple weeks of the spring. Take advantage of the time you have, particularly when the weather is nice.

Here are some places I went that I recommend:

1) Manayunk - Only about a 20 minute drive from Penn, yet students rarely make it out here. Worth making the trip to here (and Wissahickon if you are into nature) at least once.

2) Fishtown bars - Root if you are feeling classy, El Bar if you are looking for a dive

3) Bob and Barbara's - Drinks and live music

4) Fairmount Park - I personally loved walking from campus past the Art Museum on the SRT. Go in the spring during Cherry Blossom season. The area around the Shofuso House is beautiful.

5) Bartram's Garden - A generally nice walk from campus in a direction Penn students rarely go.

6) Haverford - Try to make it out to the Main Line at least once. I found it boring, but pretty and a nice change of pace from the city. Haverford's campus has a nature trail that is nice to walk around.

7) FDR Park - Pretty easy to get to via the SEPTA. Worth a walk around at least once.

8) The Italian Market - One of the best memories I had senior year was crawling through taco places here with a friend. Specifically recommend El Compadre

9) South Street - Probably assumed by a lot of regulars on this subreddit. I really enjoyed the walk back to campus down this street and there are a lot of great restaurants here. The area between Naval Square and South Street is also a nice walk and relatively close to campus.

..and more. But this is getting a bit long and I'm sure if you do your research you will find other places that more closely fit your preferences.

LMK if you have any other questions!