Great advice so far.
Precautionary measures: Definitely water. In the summer (if it's cool enough for my heavily coated dog to go hiking) it can be as much as a gallon - just for the dog.
Equipment specific to dog: Comb, tick remover, booties, vetrap, scissors (for trimming fur away from wound), pliers & wire cutters (for an off lead dog), spare leash, spare collar, muzzle (an injured dog may lash out), blanket in winter. In addition, for overnights, a closed cell foam sleeping pad, and a blanket, and of course kibbles. Rabies certificate. In some states, the collar tag is not considered proof of vaccination. On the collar, an id tag with dog's name, your complete name, and address; dog's town license tag; rabies tag. Information sheet in my pack regarding the dog, including alternate contact, vet, health issues, in case I am incapacitated and someone must care for him (same thing attached to dog's crate in car). If off lead, I would attach the same in a small baggie to the dog's collar, in case your dog runs away and is found by someone else. You might want to consider microchipping and/or tattooing as an additional measure of identification.
Contingency plans: for people with dogs small enough, get them accustomed to being carried. That way, if they are injured, they're already used to being carried and won't protest as much. My plan for Dugan four-legs, who outweighs me, is that I could (hopefully) rig a travois or drag with a tarp and my hiking poles.
Some special concerns for sighthound type dogs - this may apply depending on how "greyhoundy" your dog is. Sighthounds have very little body fat, and thus very little natural insulation. Most people realize this means they do not tolerate cold well. However, it also means they do not tolerate heat very well either. If your dog has the thin tight fitting skin more like a greyhound, it will be much more susceptible to cuts and scrapes. In addition, the cuts are apt to rip open further since the skin is tighter.
Overheating: watch the tongue. The warmer the dog, the thinner and wider the tongue (increases surface area to enhance cooling via evaporation).
Hypothermia: given that I like the big furry dogs, this is not one I know much about.
One caution - water ice - as in ice cubes from the freezer - you may want to disallow that. Water ice is dense and brittle enough that it can crack the dog's tooth enamel.
On lead vs off is a decision you have to make according to the dog. If you don't have a good amount of control by signal or voice, then the dog should remain on lead. There are some risks with off lead travel - your dog can run off, or it can get into trouble while out of reach (injured while running fast, porcupine, skunk, tangled in barbed wire, fall into an old well, step in an old leg hold trap, etc)
Another rule I try to follow is that my dog should not interfere with other people or animals (domesticated or wild), unless the people request that he do so (as in, "may I pet your dog"). I keep him a few feet away and yield trail space whenever possible. I don't allow him to chase wild life.
As for health risks, you could ask your vet to screen your dog for heart problems, and to manipulate joints to check for stability.
Physical conditioning should be gradual. Dogs can suffer from "weekend warrior" syndrome just as much as humans. Build mileage gradually, and keep a close eye on the dog so you learn your dog's body language for if he's too warm, too cold, tired, etc. Dogs can and will work themselves to death. It's up to the humans to be smart enough to keep them healthy.
How much a dog wants to hike can depend on the dog. I used to have a Pyr that would joyously accompany on +30 mile day hikes and be raring to go for our morning run the next day. I had another that would do a sit down and refuse to budge if a walk started approaching just a couple of miles - and yet she was completely sound.
Depending on your location, you might want to vaccinate against lyme disease. It is now considered endemic throughout much of southern New England.
And yes, as Kevin, said, approach other dogs with caution regardless of what their person says. On a walk at Wachusett a couple of years ago I found myself in a horrible situation with my dog being attacked by another. Yes, this was a real attack, the other dog put holes in Dugan's neck. It took 3 people to remove the dog from mine - Dugan four-legs was very lucky that a pair of hikers came along that were willing to involve themselves in this dangerous situation. The dog was off lead in an area that requires leads. The owner ignored several requests from me to leash the dog. I led Dugan who was on lead about 20 feet off the trail to allow them to pass. And the whole time her dog was attacking mine, the other person kept saying how friendly her dog was.
Good luck out there!
