I could comment on any specific neighborhoods you might be curious about. I've lived in Jax most of my life. (Although I'm moving to Philly in the fall to go to Temple Law.)
I live in Springfield, which is an historic, urban neighborhood. Good if you like big, turn-of-the-century houses for relatively low prices/rent.
For pure pleasantness, I'd recommend Mandarin or any of the new housing in the area of Butler Blvd. and 9A. For charm and atmosphere, I'd suggest San Marco or Avondale. Or, Riverside, which is similar to Avondale, but less refined and less expensive. These last three are all "historic" neighborhoods, which is something of a novelty in Florida. Not many old houses in a state where life was miserable before central air conditioning became commonplace.
Generally, apartment rent and house prices are pretty reasonable in Jax, so you should have plenty of choices, but you're likely going to be doing a lot of driving. Jax is very "spread out." I'm sure you've read by now that the city covers an entire county and is the largest in land mass in the contiguous 48 states. And people really do drive all over that area and beyond for shopping, movies, etc.
You might think you could find cool apartments downtown, but there are very few apartments or homes of any kind downtown. The ones that you will find tend to be very new (there's a recent drive to add housing there) and very expensive, upscale places.
I'd suggest you go to the campus, then drive out from there looking for something nearby. The school really is in a pretty good spot for being "central" to the places you'll want to go in town. Honestly, I'd probably choose a place somewhere along 9A or Gate Parkway if I were moving to town to go to FCSL.