Always have sharp tools. They cut clean, even against the grain if you are careful. They are also safer to use than dull ones, because you exert more force and that is when things start to slip and slide and bite you. So, learn how to sharpen properly first. Use woods that are easy to carve so that you can learn the basic techniques of carving without having to fight the material, that would be Basswood. Get to know the mechanics of wood. Understand grain and how it can help you or fight you.
Learn to do your own designs, i.e. learn to draw a bit. You will be the envy of everyone else if you can create your own original designs. This is also something you can sell if you wish.
Start with a simple design, something that you can achieve fairly easily, to get a sense of achievement going.
Experiment with different branches of carving to find the one that you really love, e.g. chip carving, relief carving, caricatures, realistic sculpture, wildfowl, etc. There are so many. I have decided that I want to be known as a sculptor.
Remember to have fun. If you don't enjoy yourself any more doing this, then drop it, or change your approach.
How do you keep the grain from tearing away with the cut? Should you cut in from both sides?
Sharp tools is step number one, all tools don't come sharp from the shop. Going in from both sides is one solution. You could also try going across the grain. Actually, this is the easiest way of removing a lot of wood fast. The best is to try and always cut with the grain, so going if from both ends if you want a really clean cut is the solution. Cutting across the grain (90 degrees to the grain) will stop the tearing and be very easy, but you will not get such a clean cut as with the grain. By clean I mean "shining", so that you won't have to sand it before finishing.
I did not have many tools, in fact it was a set of twelve small woodcarving knives, more than chisels, that you buy for a couple of bucks at any hardware or art store. So you don't need a great set of tools for this.
One more note regarding tools, I did have access to a router to take away the background, but that just makes your job easier and quicker. It is not essential.
A word on Safe Carving: