I'll start with trimming nails because it is the subject that I receive the most questions.
Yes, dogs and cats need their nails trimmed - just like humans and horses. All toenails grow and if left untrimmed they can have problems such as curling as in the photo above. This can make it difficult for animals to walk.
I suggest calling a professional - called a blacksmith or farrier - for a horse, but owners can do their cat or dog as long as they have the proper equipment and knowledge. And as long as the dog/cat is agreeable!
Pedicure devices sand down the nail or you can find nail clippers for pets. It really depends on what your pet prefers. I've found that small pets are often scared of my noisy electric sander called a drummel. So a small clipper works best and I'll file the rough edges with an emory board. Larger dogs seem fine with the drummel, but if they are worried about feel or noise, I'll use a larger clipper.
BEWARE OF THEIR BLOOD LINE! If the nails are light colored, it is easy to see the blood line and just file or cut above it. Nothing makes a dog hate to have its nails done if the blood line is hit and the dog feels pain. If the nails are dark, go slowly just taking off a little nail at a time. Just because the nail has curled doesn't mean that is where you should cut.
One way to tell the correct length is to have the dog stand on a flat, hard surface. The nails should be just above, or barely touching, the surface.
For cats, just clip the pointy tips.
How often? Some dogs require clipping every month, while others are fine for two to three months.