Banking - Schwab Online doesn't have ATM fees or foreign transaction fees on withdrawals. JP Morgan Chase may have the same. Some students open local accounts, but this can be a hassle as getting debit cards takes a few weeks. Others rely upon their home accounts and use the ATM, but this can involve fees.
Cell phones - the two major companies are Digicel and bMobile. Apart from the frequencies there is not much difference between the two. If your cell phone is unlocked (if you call AT&T and have been a wireless customer for three months, they will send you instructions on how to do that), you can use whatever SIM card works. I think bMobile is 850/1900 and Digicel is 900/1800. If you have a quad-band phone then you are in the free and clear in regards to compatibility.
I would recommend against bringing an uber-expensive phone but people do and that's their choice. While having a full-QWERTY keyboard is good for texting (which is very cheap compared to calling), if your phone gets ruined you're out more money than having ruined a 16 USD Coral 100.
However, if you come down and buy a phone, the Coral 100, while otherwise very good for a basic phone, does not have a vibrate motor.