Veteran Mode is unlocked after completing Rookie Mode. They are the exact same missions that you played during your Rookie Mode, just a little more difficult, obviously. By the end of Rookie Mode you should have unlocked most of the items in the armory. These will carry over to your Veteran Mode playthrough to make it a lot easier for you.
The items that I unlocked in the armory first were the LMG (Last Weapon), and all of the extra health slots. Don’t worry about the extra ammo magazines or the grenade slots until after you unlock the health and weapons. Remember that you can always redo how you have unlocked everything.
During this playthrough you may have some trouble on the parts where you are in control of the machine gun turret. Reason is that you must look all directions in order to kill everyone. There is no cover at all so you will get hit A LOT. Don’t worry about getting killed as you will most likely restart at the same spot where you died, with a little bit more health to give it another attempt. The loading screens after getting killed are almost instant so just keep trying and you’ll eventually get past it.