Hi Kimberly, I have also run into the same problem, the problem being that this phone is not well put together. When you slide the phone open and closed fast or sharply, the bottom of the screen-half of the phone pairs up with the keyboard to apply large amounts of friction and pull upon the LCD cable. Eventually, this (thin, wide, and black) cable begins to tear at the folding point. This means that when you slide the phone open, roughly half the cable will disconnect, so the screen gets enough power to turn on, but not enough power (or commands) to activate any polarization in the display. This probably means that your touchscreen stopped responding as well, so that you really can't transfer any of your stored data. If you take it to Verizon, they'll almost definitely tell you to get a new phone, and that they won't be able to recover anything.
If you want to save your data, my suggestion is to get another Brightside (probably $60 or less), and follow this guide (Samsung Brightside Logic board Replacement) to remove both your logic board and the logic board of the new Brightside, then simply swap your logic board into the new Brightside. If you put your new logic board in the old phone as well, and plug it into a charger, you can always keep one battery in there at all times, swapping batteries when you're out of charge.
TL;DR: Your LCD cable is broken. Either
1) Put your logic board into a new phone
(Samsung Brightside Logic board Replacement)
2) Find a new Brightside screen and transplant it onto your old phone (MUCH MORE WORK)
(Samsung Brightside LCD Replacement)