Introducing yourself in Italian is one of the first things you will need to learn in order to communicate and socialize. Although this is a fairly basic phrase, non-natives will often make the mistake of translating from English literally and using the expression 'Mio nome e... followed by their name.
Whilst there is nothing grammatically incorrect about presenting yourself this way, you do run the risk of sounding like a game show contestant.
A more natural expression is 'Mi chiamo...', pronounced [mee kya-mo], and meaning 'I'm called...'
Therefore, if your name was Elizabeth, you would simply say 'Mi chiamo Elizabeth'.
If you were looking to immerse yourself fully in the Italian language, you could even use the equivalent of your name in Italian. For Elizabeth it would be Elisabetta. To find the equivalent of your name in Italian why not ask a question on Blurtit and see if an Italian-speaking member of the community manages to answer you (well, before I do anyway!)