Using a shaving brush is without a doubt the best and fastest way to improve a lot your shaving quality. This is because using a shaving brush exfoliates your skin, making it softer and less subject to burns, nick and cuts, while at the same time putting the hair on your face straight up, making it more simple to cut. Also, the thin film of soap on your skin lubricates it extremely well (most of the time through glycerine), letting the blade you are using run over your skin easily and smoothly, improving not only the shaving quality but also making the shave a lot faster, because of the less passes needed.
But to actually get all of those benefits from a shaving brush and shaving soap, you need to use them in the right way.
First, you need lots of hot water. It must be hot and not tepid because you want your skin to be the softest and your pores dilated. Splash your face multiple times, soaking well your beard. The wetter, the easier to cut. If you feel your skin is a little dry, try applying a little hydrating cream to prevent damage to your skin.
After that, get your soap and brush. You also need a cup or something similar to create the foam you will use. Soak the brush in the hot water, then pull it out, letting the excess water drip out. It is important to use hot water because it dilates the bristles and makes the brushing a lot more pleasurable. Then, if you are using a solid shaving soap, rub your brush against it a few times to get the right amount of soap out of it. If you use a shaving cream instead a pea sized part of cream in your cup is more than enough for your shaves.
Then, start moving the brush against the cup in circular motions. You want to air to be incorporated in the foam to make it soft and consistent. Continue the motion until the foam looks like meringue (it may take, depending on your brush quality, up to 3 minutes). You then have your shaving foam to use.
Final foam's quality really depends on the quality of the products you are using: a boar brush will make a lot more foam and higher quality one from less soap, but of course it is far more expensive. But do not worry, you can obtain satisfactory results even with a $10 brush.