If the Zombies come, you are going to need a well-stocked First Aid kit. If you are on the run, you are going to get scraped and scratched. You're even going to skin some knees.
|A Zombie Mob in London - photo by http://www.cgpgrey.com [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons
At a minimum, you are going to want to have plenty of bandages of various shapes and sizes, gauze, bandage tape, and lots of betadine, hydrogen peroxide, neosporin, and the like. You'll also want tweezers for removing splinters.
Beyond the bumps and bruises you may need to deal with twisted ankles and even broken bones. Those sorts of serious medical issues require real first aid and medical training. They are the sorts of thing to learn about and train for now. You can get training through the Red Cross.
Finally, you may have to deal with Zombie bites. Let's just hope the betadine works!