I asked myself the same question a couple years ago when I started hiking with my dog, my main concern at that time was my dog getting bitten by a rattlesnake.
So I asked a friend who is a veterinarian and he told me the most important thing was knowing how to respond to an emergency, especially how to immobilize and carry your dog out of the wild in the event of a broken leg, snake bite or serious injurie. Be sure to carry some kind of strong cord, rope or tape to improvise somekind of stretcher.
Always have an emergency number to your veterinarian and be sure to know your dog, so you can actually notice if it is limping or if it seems in pain. Learn to examine your dog, check under the paws for thorns, any swelling on the legs, always keep it hydrated during a hike. Remember dogs get a little defensive under pain so be really careful.
Other than that, whatever you carry on your first aid kit will be useful for your dog too (only in larger quantitites depending on the size of your four-legged fella), dressing, gauze, bandages, micropore, iodine solution. Be careful about medications, though. Some human medications will work with your dog, however dosis are very different. Be sure to ask your veterinarian about any pain killers or other medications before giving them to your dog.
One practical recommendation, carry some nice tweezers to remove any thorns, if your dog hikes on rough terrain, you may wanna try some fancy hiking boots or be sure to carry moleskin and enough gauze and tape 'cause the cushions undes the paws always get scuffed up. If your dog has a black coat, keep it cool during a daylight hike using a slightly wet white cloth.
Dogs love the outdoors but they're too curious, so always keep an eye on them.