I'd avoid downtown at night - it is desolate.
The Fondren area is very artsy and fun. Brent's Drugs (
www.brentsdrugs.com) - has a great soda fountain/food. It was used in the movie "The Help". Other fun stores as well. The Belhaven neighborhood is very neat if you have time to just drive around - Eudora Welty's house is there. It's right near the Fondren neighborhood. It's just residential but a very old area so there are neat houses to see.
One thing to note - Jackson has a decently high crime rate, even in good areas, which is why many of my friends have left for Madison/Ridgeland. I have relatives still in Jackson in the nicer areas, but they have been daytime robberies (at open air shopping areas like Highland Village), not sure if it's still going on but it has been an issue in the past, as my friends/relatives have noted. This isn't to scare you, but to make you aware.