Right outside of downtown (and in addition to marleyisdead's recommendations, which are all quite good), I would recommend Matthews (in San Marco) and Orsay (in Avondale). Both are less than 10 minutes from downtown.
Matthews is probably Jacksonville's best reviewed restaurant. Its an excellent option if you are looking for fine dining.
Orsay is owned by the same owner as Chew (which marleyisdead recommended). Both restaurants are excellent. Orsay has a semi-relaxed atmosphere, and the cuisine is sort of French bistro style (not a perfect description, but the closest I can get). If you like shellfish, it also has an excellent raw bar. There is also a creative drink list. For the money and quality, Orsay is probably my best recommendation.
3630 Park Street, Jacksonville, FL 32204
2107 Hendricks Ave., Jacksonville, FL 32207
I just want to mention that for New Year's Eve. you should decide where you want to eat and MAKE RESERVATIONS NOW! Last year, my sister and her crew came down from SC on New Year's Eve. She called me from GA to make reservations at PF Chang's or Maggiano's at St. John's Town Center. Both were booked to capacity with a 3-hour wait for standby. We ended up eating at a Golden Corral, of all places--the only place nearby with available seating.
Running from bullets -
For nice local places to eat - BBQ is all around town,
Downtown there is a Ruth Chris - don't recommend it for the money, you get a better steak in Mandarin at The Tree Steakhouse on San Jose Blvd.
There is a III Forks (3 forks) just opened up - very nice restaurant at Tinsel Town.
For good wings - Ken's Winghouse at Tinsel Town
For Mexican - Ole is just on north main street just north of State & Union - very good mexican food
Local seafood - will get back with you on that one, everyone gets most of there food from the same place so it depends on the style.