The beaches is the next area of town. It's a totally different lifestyle and mentality than the rest of Jax. A lot more local restaurants and stores. Very laid back. Lots of bars and a party scene in Jax beach. Neptune beach and Atlantic beach are more family oriented. Probably the second best schools in town. Far away from most jobs, though.
Next is the Westside. It's mainly a suburban area with older neighborhoods. If you like houses with bigger lots you will find more in this area. It depends on where you are, some parts are crime areas and some parts are nice while other parts are slightly rural/country. Riverside and 5 points which are near downtown are really cool, artsy areas ( a lot of antique stores, hippie clothing stores and coffee shops, art galleries), also right on the St Johns river. Avondale is also a nice areaon the Westside. Jacksonville Heights is more on the country side, it's growing though and starting to lose a little of that feel. Schools on the Westside, for the most part are bad.
Northside is very unpopulated and undeveloped in comparison to the rest of Jacksonville. This area let's you know that you're still in the south. A lot of farms and local stores/restaurants. There is a nice new towncenter near the airport that brings some convenience to this side of town. It is slowly starting to dewvelope in recent years and there are some nice new neighborhoods popping up. Traffic is by far the best in Jax. Schools are not great but not awful either. Crime is not that bad ( often people confuse north downtown with the northside, don't listen to that it's not the same thing). There's a lot of industrial jobs in this area.
Last there is downtown. Currently, there is a lot of crime, the worst schools and bad traffic during rush hour. It is still convenient to jobs downtown and in the southside. Hardly any shopping at all. There are some cool things like festivals, parades, boat parade, fireworks, concerts and sporting events downtown. They are starting to clean up and restore old neighborhoods and have built some nice condos and aprtments. If this keeps up downtown will be a decent spot to live in 10 years. Hope this helps!