I lived in Jackson until 2003. Places I enjoyed include:
Nick's: fresh fish and the crab/brie soup
Bill's Greek Tavern: God Bless him, America & his fresh seafood
HUGE jalapeno burger at Stamps, the place just south of Bill's
Elite: but only for the great corned beef & cabbage served with awful canned potatoes
Thai House: Friday? lunch when Tom Kha is on the buffet
Parker House: love the steak, rolls, wine list & coffee & dessert by the pool, but the side dishes always disappointed
Bully's: for the best soul food in town
Rossini's: great Italian, great decor & service

For fine dining try Nick's, Parker House, Rossini or maybe Shimmel's