One of the first things you learn in a new language is pronouns – I, you, he, she, and so on. Japanese makes this a bit more complicated, but here are the basics.
The most basic way to refer to yourself is watashi (私 – I, me). It means both 'I' and 'me' since this grammatical distinction isn't made in Japanese. This is a perfectly good way to refer to yourself in any situation, no matter how formal.
If you want to make it more formal, you can use watakushi (also written as 私). This is used only in formal situations.
There are a few feminine ways to say 'I.' One is atashi (あたし). Leaving off the 'w' sound at the beginning is commonly done by women.
Uchi (内) is also used, usually by adolescent girls. I've heard girls as young as about 5th or 6th grade use it.
Boys and men often refer to themselves as boku (僕). This is perfectly okay for formal situations, but only used by males. I've been told it's a good way to refer to yourself in the workplace when talking to your coworkers.
A rougher word used only by boys and men is ore (俺). This is definitely not formal and should only be used in casual situations, like among friends or drinking buddies.
There are two other male versions which are good to know, but you really shouldn't use. One is oira (おいら), which as far as I know is only used by anime and manga characters, or by boys who are too into anime or manga. Apparently this is also an antiquated form that was used by monks.
The other is sessha (拙者) which was used by samurai. It's best not to use this form unless you're trying to convince people you're a samurai.
Two forms of the first person are used often in business settings. These are kochira (こちら – this way) and ware ware (我々 – we), which is sometimes shortened to wa (我). A rude version of kochira is kocchi (こっち). It's not important to know these for everyday conversation
Finally, another option is to use jibun (自分) which means something like 'myself.'
But here's something important to know – Japanese people don't often refer to themselves much. The pronoun is a grammatical necessity in English. Even if you know who is doing what in the sentence, you still need to say it. Not so in Japanese. The subject is implied whenever possible. In fact, it's best to use it only when necessary. Think of it as saying, 'Well, in my case…'
If, as you would in English, you start every sentence with a reference to yourself, you either sound like an egomaniac or a student of Japanese who is still at the level where you don't understand how its pronouns work.