Brett's right...from the beach is nearly an hour...and alot more if traffic is sketchy.
Southside seems to be the general consensus for single folks especially. The deal is that it is a 20 ish minute ride to the beach bars and around a 30 minute door to door to work. There are a million big bix mart stores, restaurants etc...in the area. Look in the vicinity of Gate Parkway, Butler Blvd, St. John's Town Center etc...This is the "trendy" place for studs to live.
Orange Park is, in theory the short commute. However, while my buddies in Southside were driving 30 minutes and 20 ish miles each way, I was driving 8 miles in the same 30 minutes. Orange Park is not a good deal because of its "proximity" to base. There are reasons to live there...if you really want a house vice an apartment or townhome for example, Orange Park is your best bet. If you are concerned about schools, etc...there are reasons to go there. It has the same assortment of big box marts, with the exception of the high end stuff at St. John's Town Center, which is a 40 minute drive. The trouble is...as always in Jax...traffic. If you live in Argyle Forest, the main development there (huge...probably 5000 houses), you can plan on a good 10 or so minutes just getting out of the development.
The other real choice nearby NAS is Doctor's Inlet. There are some nice single family houses down there, it is close in terms of mileage to NAS and if you are looking to get away from big suburbia, its probably your best bet. I never made the drive to work from there, but it should be slightly shorter than Orange Park proper. There are some small-ish shopping centers down there with the basics, Target etc...but the area is growing. If I had it to do over again, I would probably take a look down that way, as I needed a house.
Overall big picture take-away about Jax though...is its damn big. You'd better get used to spending some time in the car to go anywhere and wherever you live, you will be at least an hour away from some other place you want to be! Good luck and feel free to hit me up with any specific questions.