Next is a greyhound comb. After brushing your dog with a slicker brush, go over the entire dog with a metal comb. A greyhound comb will catch any knots or mats you've missed with the slicker. You can then go back with the slicker and re-brush that area or work the knot out with the comb. Again, be sure to comb all the way down to the skin to avoid combing over any mats or tangles.
My favorite tool to use on short haired dogs is a Zoom Groom. It's a rubber brush that removes shedding hair from short haired dogs. Basically, you just brush the entire dog with it and all the loose undercoat clings to the brush. It's a very fast way to reduce shedding in between baths.
Dematting tools help to cut through and break up mats. It is important to not let the blades catch on the dog's skin, but as long as you are careful and make sure you are only combing through hair, they are safe to use. Dematting tools cut mats into smaller chunks, which can then be brushed out with a slicker or comb.
Scissor-style nail trimmers (like the ones pictured) are the safest and easiest to use. Dogs' nails should usually be trimmed once a month. If they are overly grown out, they may need to be trimmed once every week or two. There is a vein that runs down the middle of each nail, called the quick. As a nail grows out, the vein grows with it. If nails are trimmed more frequently, the quick will slowly recede, allowing the nail to be cut shorter.
A dremel can be used to file down nails. Using nail clippers alone leaves the nails rough and jagged. The dremel smooths out the nail and also gets the nail shorter than just clipping.