A lot of books have been written about stalking and it makes no sense to reinvent them. Also, I will not claim to be an authority. Tom Brown Jr has an excellent series of books about stalking and other basic survival skills from a Native American viewpoint. I highly recommend them.
I will give you the most basic steps to get you started till you can get one of the books or find an expert to work with.
Most animals watch for motion. If an object doesn't move, they take it to be a rock or tree or something similar. Stationary objects are generally not seen as a threat. Even slow, fluid motions, such as a branch in the wind are not seen as threatening.
You're probably not going to hide from a deer, but if he doesn't see you moving and he doesn't smell you, he will generally ignore that strange-looking tree that wasn't there yesterday (you).
When stalking, only move when the animal looks away from you, and do it slowly, smoothly, and quietly.
When you put your foot down, test the ground first. Don't put weight on that foot until you know there's no sticks or other objects under it that might snap or make you lose your balance. If there is something, carefully move the foot and try again.
Some folks would say an animal will sense your aggressive thoughts, some would say your aggressive thoughts increase your nervousness and affect your motions, and some would say both are true. Bottom line, if you cultivate calming, peaceful thoughts, you will have better results when stalking animals. It might help to imagine gently petting the animal.
Try to avoid excessive clothing that may make swishing noises when rubbing together or against objects. Ideally, wear as little as weather or other conditions will allow. A breechcloth and soft moccasins are perfect stalking clothing, but maybe not so much in Winter or in a thorn thicket.
If you need to be warmly dressed, try to wear wool or leather clothing. These tend to be much quieter. Avoid solid colrs when you can. Patterns tend to break up your silhouette. Why look like a strange tree, when you can blend with the background? Camoflauge patterns are obviously good, but even a plaid shirt does well. If you only have solid colors, use earth tones. (Tan skin blends surprisingly well with a wide range of backgrounds.)
Regardless of your clothing or lack thereof, be aware of any branches you might snag on that can give you away.
Animals use their sense of smell as much as their sight or hearing. Always stalk downwind or crosswind. Stalking upwind of an animal nearly guarantees it will alarm and take off. (The deer might have ignored the strange tree, but not if it smells funny.)
You might have some luck with masking your scent, but don't count on it. It's much more reliable to just make sure the wind is in your favor. This may mean traveling quite a ways around before getting close to your animal. Be sure to keep the noise down while you're doing that.
Be patient. It could take an hour or more to get really close, but it's worth it. Being an arm's reach from an animal who doesn't know you're there is quite an experience.
For hunting purposes, you don't generally need to get that close of course. In extreme survival cases though, you might have to take an animal with your bare hands.
But having good stalking skills is rewarding in itself. You'll just have to experience it yourself.
While humans do notice stationary things, you can still play a stalking game to sharpen your skills. One person plays the animal while one or more others stalk. The animal may look around, and if he sees the stalker(s) move, the stalker(s) must start over again. If the stalker can touch the animal without being spotted, he wins.
Imagine the thrill of touching a real wild animal! (Of course this is not wise with skunks, bears, and some others. Use good judgement.