My in-laws are from there, and whenever we go visit we spend a lot of time at the beaches -- but not Jax beach (usually Vilano). Might be a bit cool for that in November, so...
Castillo de San Marcos is in St Augustine and is worth checking out. There's also a lot of small shopping places in the
old city part just across the yard from the fort. I remember it being quite touristy, but that was a few years ago.
There's lots of shopping and restaurants at
the Jacksonville landing.
The treaty oak is really something to see. Absolutely HUGE sprawling oak tree. Sounds boring, but it's pretty amazing.
If you're into board games,
CoolStuffInc has a store in Jacksonville.
Grab breakfast
at the Metro Diner. They've added more locations since being on DDD, but in my experience the quality hasn't really suffered -- more locations just means shorter wait times. If you like sweet breakfasts, Yo Hala on the Square is probably my favorite sweet breakfast from anywhere.
The Jacksonville Zoo is pretty cool if that's your thing. I don't remember anything amazing about it, but I do remember thinking it was one of the better zoos I'd been to.