"Hello. My name is ____. Nice to meet you."You just say
"Ang pangalan ko ay ____. Kinalulugod kong makilala kayo."The last sentence is actually not necessary. I think it is understood, more or less, if you are indeed glad to meet the other person or not. It means "I'm happy to meet you." Alternatively, you can say "Ako'y nagagalak na makilala kayo," which means "I'm glad to meet you." Or, "Mabuting nakilala ko kayo," which literally means, "It's good that I met you."
You can say "Hello," "Hi," or "Hey" and it would be more or less understood by most. The equivalent of Good Morning, Good Afternoon, Good Evening, and so on is generally used to say hello.
"Kayo" is actually a plural of "ka," which both mean "you" in this context, but the plural form is used for politeness. Otherwise, "Ako'y nagagalak na makilala ka," and so on is sufficient. Likewise, no need to say the whole "My name is," just an "I am ____" will suffice in most occasions, which translates to:
"Ako si ____."A shakehand will do. But it is not a very common practice so expect a weak handshake on occasion.
And lastly, when introducing yourself, given names are enough. If you should mention your full name, you can mention given name then family name.