There are almost as many ways to cut up ingredients as there are ingredients to cut up, but the basics of chopping vegetables and meat are fairly straightforward. As with any task, if you practice slowly, speed will come in time - do not try and speed-chop like you see on TV straight away; you'll just cut yourself, and there will be no prizes for finishing first in any case.
A good, sharp knife is a must. Somewhat counter-intuitively, a sharp knife is usually safer than a semi-blunt one because it is less likely to slip off whatever you're chopping and bury itself in your hand or finger.
Invest in a decent knife early and you will get better results and enjoy the process of preparing ingredients so much more. You don't have to spend a fortune and a decent knife from the likes of Global or Wusthof for $150 (Australian) will last you for many years more than a cheap, quickly blunted $30 knife from the supermarket.
You need perhaps 3 knives at the most. Knife manufacturers have a knife for every occasion, but you only really need a good size chef's knife (which will do everything from jointing a chicken to mincing garlic), a small paring knife for very delicate work, and a serrated knife for bread etc. To start with, the chef's knife is all you need.
Finally, an excellent source of chopping how-tos is YouTube. Simply search for 'how to cut vegetables' (or search for how to chop a specific item) and watch away. Buy yourself a bag of cheap onions, carrots, and/or peppers and practice away - you can always portion and freeze the veg ready for use later.
Good luck, have fun, and remember we are here to answer all your cooking questions.