As Kimberly has mentioned, there are many options right outside the city that will fulfill your needs.
What school will you be attending? Widener?
If you are looking to remain in the city then South Philly or SW Philly are really your best options when it comes to travel time. They are both super close to I-95 and Chester is just a hop, skip, and a jump away! :)
I personally moved to PA from NY years ago and have lived in both the city and the suburbs. You can certainly get more of a backyard on the outskirts of the city and a bigger space for your budget of $1200. It is difficult to find a space in that price range that will also accept a dog AND have a backyard space in the areas of the city that would give you the 30 minute commute you desire. They do exist, but those are the types of properties you need to jump on right away to make it your own.
SEPTA and the Regional Rail service are great options for travel depending where in Chester you need to go and when you will be traveling there. There is a bus or train for most places around here.
If you are looking to move to the area at the end of August, the best time to secure a rental is between June & July. During those months, there will be more on the market because many leases end during that time and landlords are looking for new tenants.
The rental market in Philadelphia moves very quickly. When you visit, make sure you get all of the available listings the week before and your Realtor should keep an eye on the properties to ensure they are available to see when you arrive for your showings.
If you have any questions or need some further guidance, I would love to hear from you. Feel free to give me a call or send an e-mail at anytime! :)
Best of luck with your move and welcome to Philadelphia!
Century 21 Alliance
3012 Township Line Road
Drexel Hill, PA 19026
610) 449-6006 (Main Office)
(610) 449-0645 (Fax)
(914) 469-7672 (Cell)
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Other good options would be rentals in places like Darby, Ridley Park, Essington, Prospect Park, and other suburbs just south of Philly proper. They're all within fairly easy distance of train/regional rail lines if you wanted to take public transportation into the city, are a very easy distance from the highways that go to Chester, and would likely be easier places in which to find a lovely, affordable place with a private backyard. :)
If I can help you in any way, please let me know! My contact information is below, and I hope to hear from you soon!
Real Estate Agent: Keller Williams - Center City Realty
REALTOR® - Member of GPAR
Cell: (267) 530-5641
Office: (215) 627-3500