Make sure you own a good sturdy pair of boots, preferably made of leather or ballistic nylon. Take care of your feet and they’ll keep you running.
Get a large bladed implement. Machetes, axes, swords and the like don’t run out of ammo. They’re also useful for clearing brush, making shelters, etc.
Don’t forget your books. There’s no TV after an apocalypse and no power for PCs or other electrical devices. You can learn medicine, agriculture, engineering and other life saving skills with books.
Keep two sets of clothing with you. Preferably natural fibers, something that breathes and cleans easily. You won’t be able to go to the dry cleaners.
A good heavy jacket or coat with pockets. Pockets make it easier to carry small supplies, and a good heavy coat can double as a blanket. It will get cold out there so bundle up.
If you can choose a vehicle, try to use something with plenty of space and good gas mileage. A brand new Camaro looks cool, but an old pickup truck is far more useful. My personal recommendation would be an older two and a half ton military truck also known as a “Deuce and a Half”. They run on diesel which means you can use biodiesel otherwise known as old fry oil that’s been filtered. They are also tough as nails and are built for almost any terrain.