Typing speed is something which most of us don't really care about in the beginning. But when we see someone typing really fast on the computer keyboard, we feel that 'I wish I could type like this person'. The average typing speed of any normal English speaking person is most likely to be between 30-35 words per minute which is decent enough. After all accuracy also matters a lot. You can find out your own typing speed by indulging in some online typing tests which are pretty much accurate. They give you both - words per minute and accuracy.
If you don't score well, I am sure that you will wish to type on the keyboard without looking at it. Imagine sitting in the dark and chatting with someone on IM, will you be able to do it? That's the question you should ponder about. To start with, make a mental map of the keyboard's keys. The QWERTY keyboard layout is the most used layout. The first tip is to avoid looking at the keys and start typing. It will be very difficult in the beginning but you will catch on.
Always maintain proper posture while typing. Do not slouch. You should have a proper arm rest and both your hands should be parallel to the ground when you type on the keyboard. Fingers should be slightly raised up while typing. If you can buy an ergonomic keyboard, it will help you better. The keys are arranged in such a way that it helps you type better and also avoid carpal tunnel syndrome problem.
Initially when you practice, do not worry about the mistakes you are making because that is part of the learning process. Instead focus on the task on hand. If you are copying some handwritten text to the computer, read sentences at a time instead of individual words. This will also help you type faster as you don't have to see the text again and again.
Finally, take some online typing tests which are specifically designed to help you improve your wpm count. Fast typing is a skill which develops with sufficient practice and experience. All the best.