To train your new kitten, you need an understanding of how she thinks and behaves.
Cats are not like dogs. Dogs are pack animals who want to please their 'leader' (you). Your kitten will be more interested in pleasing herself!
That said, with a bit of understanding and patience, there's no reason why you can't train your kitten. And it's important that you do. After all, how will you groom your kitten or keep a close eye on her health if you don't train her to be handled? And do you really want her padding about on your kitchen counters?
The basic principles
The first thing to realize is that cats do not understand punishment. Encouraging good behavior is much more effective. Sound tricky? Just remember the three 'R's' - respect, reinforcement and rewards.
Let's take respect. It's important that you behave in a 'cat friendly' manner. Did you know for example that cats dislike being stared at, or that they can't abide sudden noises or movements?
Reinforcement just means consistency and repetition. If your kitten does something you wish to discourage (like jumping on the kitchen counters), always gently and firmly say no. If she does something you want to encourage, be sure to always make a big fuss of her.
Which brings us neatly on to rewards. There are two types of rewards - praise from you and a tasty treat. Both are very motivating to your kitten.
Training your kitten to be handled
Most cats don't like being picked up, and the earlier you get your kitten used to it the better.
A lot of people unwittingly teach their kitten bad habits. They pick their kitten up, she struggles and they immediately put her down again. Thus, the kitten learns that if she struggles, she gets her own way.
A better approach is to pick your kitten up and then, if she struggles, continue to hold her gently but firmly. When she settles, lavish lots of praise on her and set her down.
Can you train your kitten not to scratch? No. Kittens love to scratch - it's a natural form of territorial marking that also gives the muscles a good stretch. Does this mean you have to accept ruined furniture? No. You just have to train your kitten to re-direct her scratching.
Buy your kitten a scratching post (rough surfaces are particularly appealing so you might want to choose one covered with something like rope). Play with your kitten near the scratching post and then, when she uses it, reward her with praise and maybe a treat.
If she does scratch an item of furniture, she'll mark it with her scent and cleaning it with an anti-odor product may discourage her from going back there. Some people find putting something like polythene on the item of furniture helpful as kittens generally aren't so keen to scratch slippery surfaces.
Play biting is natural kitten behavior. If your kitten bites your hand during play, end the session immediately. Whatever you do, don't jerk your hand away. This just makes the game even more fun! Toys and balls are safer objects of prey.
Clicker training is a modern, kind and scientific way of training animals. You use all the same principles already discussed but your kitten also gets a 'click' to mark any good behavior. More information on clicker training.