Okay I recently started running and I was fairly fit when i started, and jumped in starting at 6km runs in the morning. Now problem came when i got shin splints. After much research I found the best way to start, no matter how fit you are, is to do power walking as well as running.
The reason is this. When you jog you build up muscle at the back of your calves. You foot lands in a way which slightly twists it, which is fine unless you are doing repeated pounding for many miles. Having stronger calf muscles tend to make that more prominent and hence the shin splints.
If you power walk, or walk on your heels with your toes lifted lots, you strengthen the muscles at the front of your shins, which balances the leg more so less impact and less injuries.
Get some decent shoes off the bat, and try to run on the grassy verge rather than the concrete also helps avoid things like this. Trails are also excellent.
Increase the distance by 10% each week only, this can be painstakingly slow at first, but its worth it.
Try the couch to 5k program for a reasonable starter, it has walking/running combos to start you off.