Welcome to the Cat Forum!! We hope you'll stick around to share your cat adventures with us, and your cat photos!
Everything you described is perfectly normal for a cat interacting with another cat. People are not cats, and they have to learn that biting and scratching, while acceptable play behavior between two cats, is not acceptable with people.
The most effective way to discourage unwanted behavior is to refrain from encouraging it. Cats desire attention, and when they do something to get attention, they will do it again. When your cat bites, say "No!" in a firm strong voice, and immediately stop interacting with the cat. Gently remove your hand or other body part your cat was biting. When the cat stops biting, then reward with praise and petting.
It seems to me that cats have three bites: play bites, warning bites, and aggressive/defensive bites. Only the last of the three is dangerous to human body parts. Learn to distinguish.
Playing with the cat is best done with interactive cat toys, rather than directly with human hands, as you would a dog. Toys on a string or a wand take your hands away from temptation for the cat.
Redirection, as you mentioned is also effective. Particularly with respect to scratching. Cats do need to scratch. Redirect their scratching to a cat scratching post, pad, cat tree or whatever.
There are many good books and articles online for you to educate yourself, as well as browsing through our postings here. This is a particularly good site for information:
Again, welcome, and tell Kahn Rocket says "Hi!"