There's nothing like running around for half the day with little bits of toilet paper stuck to your face (for the record, there are better ways of stopping the bleeding) as a result of a nick or cut which occurred while shaving. Fortunately, preventing nicks and cuts while shaving is fairly easy. I've got two words which will likely solve your problem. Be careful.
Well, yeah, that seems like a no-brainer all too simple solution, but the fact is that most nicks and cuts while shaving are caused by simply being careless. So, slow down and pay attention during your shave -- those extra few minutes will make a difference. While shaving, always use proper shaving techniques -- always use a sharp blade and high quality shaving cream and stretch the skin tight with your free hand. If you find yourself hacking your face up on a regular basis, you may even wish to consider going with an electric razor.
If you do cut yourself, you can stop the bleeding by applying a bit of pressure using a clean cloth. As with any cut, always make sure to clean the wound properly with soap and water and follow up with a bit of antiseptic to help prevent infection. Don't pick at the cut -- this will only prolong the bleeding and increase the risk of infection. To stop bleeding quickly, use a styptic pencil or powder. I like the powder better as I feel they're more sanitary than a pencil. Simply dab a tiny bit of powder on the cut using a cotton ball -- it will sting a bit, but the bleeding will stop almost instantly. I like Nick Relief Styptic Powder for this purpose.
If the cut does become infected, you can often help heal the cut by using an over-the-counter antibiotic cream. Of course, if the bleeding won't stop after 15 or 20 minutes or if it is deep, long, or becomes infected, you may wish to seek medical attention.