To start off, always greet people the first time your meeting them when you start your introduction with "はじめまして" (pronounced as hajimemashte*). It literally means "to start off", "to begin" or most of the time "first time". It should be followed by a normal vow ( will teach you later the different types of vowing for different circumstances). Remember, when you are greeted always response nicely. If you are greeted "はじめまして", respond immediately with the same greeting.
After greeting "はじめまして", you may follow it up with daily あいさつ like お早う御座います (ohayou gozaimas*)、今日は (konnichiwa*) or even 今晩は (konbanwa*). Of course you need to consider the time. After that you can then start telling about yourself. It usually starts by telling your name, age for boys ( not necessary, specially for girls. They don't like telling their age nor being asked about it), and place where you come from. In formal meetings like business or employment, they usually mention hobbies, special skills, studies and other important details that will empress the listeners.
There are many ways to say your name in greetings. For beginners (elementary or informal introduction), you can say "わたしのなまえ は John です" (pronounced as watashinonamay wa John des*) or just simply "John です" (John des*). For advanced or formal introduction, it is appropriate to say "Doe, John と もうします” (Doe, John to moushimas*). It literally means " I am called John Doe". In Japanese context, when you mention your last name, It always comes first before your given name and in formal settings, mentioning your last name is important as people will call you with your last name as sign of respect not unless you tell them its okay or you preferred to be called on your first name.
To tell about your age, you should find out the numerical translation of your age and then just simply add ~sai (counter for age). For instance, if you are 23 y.o. you can say "にじょうさんさい です" (nijusansai des*). にじょう(niju*) is 20, さん(san*) is 3 and さい(sai) is the age counter (if you want to know what is the translation of your age in nihongo just comment below and I will respond).
People from far places usually mentions the place where they come from. You can say it through "place に から きました" (place ni kara kimashta*) for informal or normal setting and "place と いう ところ から きました" for formal settings. に is a particle for place, から(kara*) literally means "from", and きました(kimashta*) literally means "came". the alter part "place と いう ところ から きました" literally means " I came from a place called place".
To end your introduction, it is important to say "どうぞ ゆろしく おねがいします"( dozo yuroshku onegaishimas*) for formal or simply "ゆろしく" in informal setting followed by a normal vow. Most people would translate it as "nice meeting you" but it literally means "Please treat me favorably" or "please take care of me".
These are the basics of self introduction in Japanese. Practice will always make it perfect. Gambatte ne!