As far as places to stay, I would recommend the very north side (outside of the City). On I-55 north the County Line Road area is loaded with nice hotels, shopping and eating areas. The biggest mall (Northpark) is also located on County Line Road.
As others have said the Natchez Trace Parkway runs through Jackson. A google search will tell you a little about it. If you have the time, travel west to Vicksburg and follow the Mississippi river south to Natchez. You can visit some historical and beautiful antebellum homes and get some serious exposure to the history of the civil war.
As already mentioned the Mississippi Agriculture Museum is worth visiting. It is located on Lakeland Drive off of I-55 north, south of the County Line Road exit.
If you like catfish and seafood go to Berry's Seafood on Hwy. 49 south at Florence, MS about 10 miles south of Jackson. I know they are open on Friday and Saturday nights - you might call if you want to go another night. They have one of the best seafood buffets in the area. They also have a resturant in Natchez. Sorry I can't recommend other places - I just don't spend much time in Jackson any more.
Personally I would stay out of the city of Jackson - anywhere near where I-20 and I-55 cross each other. For a small city Jackson's crime rate is very high. Although a little better than 3-4 years ago when they had the highest murder rate per capita in the US they still have too many drive by shootings, drug dealings, and carjackings to suit me.
I live about 100 miles away but still have relatives in the area. When my kids were small we used to take them to the zoo and other places in Jackson and felt completely comfortable. Unfortunately it's not like that any more. As in most cities a lot of the law abiding folks have moved to the suburbs and a great many of the ones left in the inner city...well you wouldn't want them for neighbors.
I hope you enjoy your stay! /forums/images/graemlins/cool.gif /forums/images/graemlins/cool.gif /forums/images/graemlins/cool.gif
TBone