I can weigh in a little on Jax, but for more details you should look into SORBA Jax. I lived in Jax Beach and worked in Jax until a year and a half ago. The only place to ride in the city was Hanna Park. It's fun, but it closes at dark. You can sneak in through the beach side after dark, but security guards roam the park after hours. So if you work a regular 8-5, you basically have to give up riding in the winter. This sucks. There is a pretty area south of Jax Beach called Guana Reserve. It is completely flat with fire roads throughout, which are handy for putting in miles and getting out into some woods. But technical or challenging it is not.
I think there were some more riding areas in the works, especially west of town.
Traffic is pretty bad at rush hour. If you are into bike commuting, just know that Jax proper is EXTREMELY bike unfriendly, as are most big FL cities in general. Every time I rode to work I felt like I was taking my life into my own hands. The road group rides I used to go on were constantly harassed by drivers. I saw a cyclist get hit by a car one time 100 yards in front of me. Turns out the driver was 80 something AND drunk.
Despite all the negatives, I met some very cool riding friends while living there. The mtb community is small but tight. If you ride road at all be prepared for an icy reception. The roadies in Jax are very elitist and not very friendly.
I've lived in a handful of places in FL, including Vero Beach, Gainesville, Orlando, and Jacksonville. Gainesville wins by a huge margin in every category except for access to the ocean. Culture, bike-friendliness, traffic, entertainment, natural beauty, you name it. Gainesville is peppered with parks. They actually value nature there. Jax paved almost all their land a looooong time ago. The rest of the undeveloped land is mostly private property and fenced off. There were at least 60 miles of trails accessible from town without the need to step foot in a car as of a few years ago, I'm sure there are way more now. Then there's San Felasco and The Rock. And Santos is close enough for weekend trips. But there is so much riding in town that I found myself almost never going to Santos. I lived there for 7 years, and moved for a girl.
I'm out West now and spoiled by access to mountains all around me (all public land, too). But one day I might find my way back to Gville, it is just that good.