The key to starting anything is simple, just start. Stand up is no different. Find a stage, get some material together, and then perform it. That's it. Don't second guess yourself, and don't procrastinate out of fear. The only way to conquer your fear is to do it. To start stand up comedy all you really need is a little time and courage.
Now find a stage. Fortunately this is easier than you might think. There are countless open mic nights and they are all over. Sites like Badslava.com can help you find one in your area. An open mic specifically for comedians is preferable, but anything with a microphone will work. I've performed at bowling alleys and had great sets. Once you've chosen a stage, schedule the date of your performance. You shouldn't need more than a couple weeks to get an act together, and you will be more likely to do it now that you have a deadline.
Before you go on stage you need to have some jokes ready. These can be written out word for word or it can just be a list of topics you want to cover, as long as you have something. Don't try to wing it, especially your first time, you will likely freeze up. You only need about three to five minutes to start. Your whole act shouldn't take up more than a page or two. Try to have some kind of punchline about every thirty seconds or every few sentences. There's no need to segue between topics. If you are struggling, watch other comics to get the idea, but do not steal material. Also The Comedy Bible by Judy Carter can give you the basics. Your best joke should come last, and your second best should be first. Remember though, there are no rules to comedy writing. Your material can be anything, as long as you wrote it and it's funny.
Performing is the most frightening part when you start stand up comedy, but it is also the most rewarding.You'll find that making strangers laugh is addictive. Practicing your material in front of a mirror can help you prepare. There are no rules for how to dress, but your clothes should not distract from your words. Your on stage persona should be an exaggeration of yourself. Speak to the audience like your talking to a good friend, and you've been waiting all day to tell them something. If you're not getting a good reaction just keep going. Drawing attention to the lack of laughs will just make it worse.
Every comedian bombs occasionally. Use it as a learning experience. If you are doing well, make sure you give the laughter a chance to die down before you start speaking again. Usually someone at the show will shine a light to indicate you need to wrap it up. Try to end on a laugh. Don't go over your allotted time though. This can get you blacklisted. After you have finished thank the audience and exit the stage. Congratulations you're now a comedian.