If you have your own car, getting around Jacksonville isn't that bad in most areas. A notable exception is the Blanding area on the Westside - traffic there tends to be a nightmare most times but 10 times worse during rush hour.
If you don't drive, the only real alternative is JTA. The network of busses is fair and most routes run at least hourly. The major problem is if your trip requires changing buses - chances are you'll have to travel downtown to the central bus hub, plus there is no transfer system so you'll have to pay fare on both buses (the monthly bus pass was reasonably priced a few years back, however I don't know if that is still the case.) One nice thing about the buses is that most of them now carry bicycle racks on their fronts, so if you're start or end points aren't right on a bus route you can bike the rest of the way.
Travel around downtown also has a couple more alternatives - there's a monorail-like transporter, but the entire length is only a couple of miles. There's also a trolley that loops thru downtown (and I believe they recently added another across the river on the Southbank).