I'm originally from Jacksonville but moved away about 7 years ago so not current on the range information but here my opinion. There used to be a great outdoor range off SR 16 out by St Augustine but its a neighborhood now. Anyways to contribute to the topic with relevant information I never cared much for Gateway, they are over regulated and unfriendly when you bring new shooters.
Went to check out Ancient City with my father on my last visit. Seemed good online but when we got there we were told that there were no steel targets anymore since some less intelligent shooters ruined them with AP and green tip rounds. The facilities were nice, owner and his range philosophy were cool, but the lack of steel and the high cost of membership were dealbreakers in the end. Would be a cool place if you're not worried about cost.
The one by Lake City in the Osceola Natl forrest is nice. Completely free, covered line, they provide target stands and backers, but its definitely a bit of a drive. Another downside is that space its limited, more so the rifle range than pistol, so it would be a downer to make the drive and realize upon arrival that its crowded. We went on a weekday late morning and 5 other cars showed up in the 2 hours or so we were there. I can imagine that its a popular spot on the weekends.
One option that hasn't been mentioned is the Bradford sportsman's farm. Its off of SR 301, past Starke maybe halfway to Gainesville. Haven't been there in probably 5+ years so might have changed but they had a good assortment of steel on the pistol range. I'm not a skeet/trap shooter, in fact I'm not even sure what the difference is, but they have a huge course with the auto towers out there for you shotgunners. Decent facilities and they leave you alone to do your thing if you're not doing anything stupid. Only downside for me was cost. Not bad if you're just paying for your own membership but I like bringing new/inexperienced shooters with me quite regularly, last time I went there it was 25 a person and I wasn't a member anymore. 75 bucks spent for just daily range fees stings when you're paying for your guests and ammo too. Also if I recall correctly they're only open certain days, that might not be true anymore. Anyways check them out online it might be what you're looking for:
http://www.bsfshootingsports.com/
Good luck hope that was helpful. I miss Florida, the only thing Arizona has going for it is lots of public land to shoot out in the desert.
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