Don't shave against the grain. It isn't good for your face.
Here's my recommendation. Really my recommendation is to learn to use a straight razor...but I'm going to assume you aren't going to do that...but you should. If you don't get a straight razor, I recommend an old school safety razor...because modern razors cartridges are a complete rip-off, especially if you replace the blades as often as you should.
Here are two websites where you can buy blades and brushes:
http://www.shaveshop.com/
http://www.straightrazordesigns.com/
So the process:
1. Lather up your soap with your brush using warm water.
2. Wet a washcloth with hot water and wet your face with it. It will open your pores and give you a better shave.
3. Lather up
4. Shave *with* the grain first (i.e. following the direction your hair grows).
*While shaving clean the blade with after every stroke. Don't ever take a second stroke when there are little bits of hair on the blade.
*While shaving have a light touch, don't press down too hard on your skin.
5. After you have shaved with the grain, for a closer shave, next shave across the grain...at 90-degree angle to the way your hair grows.
6. Now, at this point, if you are the sort of person who has really coarse and think hair and have big problems with getting a good shave, what you do now is use your fingers and feel your face. Feel for patches where there is still hair that is being a problem. For those patches only, you can shave again *almost* against the grain...but not quite. If against the grain is 180-degrees angles to the way your hair grows, go with 170-degrees.
7. Use the hot washcloth to get all the soap off.
8. Rinse out your soap mug and rinse out your brush.
9. Wring out your washcloth and run it under really cold water. Then put the cold water on your face to close your pores.
10. Use whatever other post-shaving ritual you use...after shave, lotion, etc.
Follow-up. If you have really, really coarse hair, you should be changing your blades very regularly. One way or another you should switch up every week....maybe more if your hair is really coarse. If you have a straight razor, you wouldn't have to change blades, just strop the blade...maybe get it professionally sharped once every two years.
Note: Shaving too close can lead to ingrown hairs and skin damage, so be kind to your skin.