North Jackson is the best bet. Stay close to where kbg said in the post above. Unfortunately, crime is growing here in Jackson. There is a crime task force looking on to how we can solve this problem - I believe it's a case of Jackson growing too fast for the police force - IMHO, I believe we need a larger law enforcement force. Just my opinion.
Keep in mind that it's a more expensive on that side of town, but a lot closer to shopping, grocery, hospital, etc... You'll be close to Union University which has plenty of great higher learning ops (if you're thinking of going for a MS).
The public school system is not very strong - many people opt for private schools - the private school pop is bursting so much so that there are new private schools every year & the ones that existed had to build larger buildings! But you won't have to worry about that for a while.
BTW - we are looking to move out of Jackson & I've lived in Washington, DC, San Antonio, TX, Clovis, NM and husband grew up in Memphis - so we know/share your concern about crime!
Good luck