Looking for an apartment in West Chester is easiest in the summer months. In the summer, you will have more vacant apartments that have freed up when students graduate or head home for the summer months. You will also see this effect in December and January at the end of the first term, but to a lesser extent.
West Chester landlords like to see people who have clearly thought about the responsibility of making monthly rent payments. If you can show up with good supporting documentation, such as proof of employment or scholarship, it shows the landlords that you're treating your new home as a serious responsibility.
West Chester doesn't have any sort of public transportation system operated by the city for the general public. There is a shuttle on the University campus that transports students around campus and nearby locations. Typically though, you should have no problem navigating on foot.
Just a walk through West Chester should be fascinating for someone who loves history. For a more hands-on experience, pay a visit to the Chester County Historical Society. The city has a lot of small, locally owned shops that make for fascinating shopping. Also, because it is a college town, the bars in West Chester are excellent and plentiful. Head downtown to see the best of the city's nightlife.