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During a self-introduction in class, Jan said, "Hi my name's Jan Wayne. My friends call me Jan."
I want to write these sentences down formally, but I don't know how to.
I mean I don't know whether it is a formal way to introduce oneself to other people by saying
"Hi my name's Kitty XXX. My friends call me Kitty."
Are you sure the name isn't supposed to be Janet Wayne? It's strange to say "My friends call me Jan" when that's her actual first name. Normally that pattern is used for abbreviations or nicknames, but people would call her Jan by default.
In general, I suggest not using "My friends call me X". It's not always clear whether the person you're speaking to counts as a friend or wants to be a friend. Context, can help, but some people (like me) find it a bit intimidating. You can use "You can call me X" or "I go by X" instead. This is just my opinion; the phrase is definitely correct idiomatic English.
I don't think the first sentence is necessarily informal in the sense of being potentially rude. For instance, if you were meeting the CEO of an American company face to face, it would be fine. Still, there are some changes you can make to fit various circumstances.
When addressing a group, as in a lecture or presentation, it helps to add a few syllables to make your words easier to hear:
Hello, everyone! My name is Janet Wayne. You can call me Jan.
Alternately, you can just use the abbreviated name to begin with:
Hello, everyone! My name is Jan Wayne.
When meeting someone face to face, you can add an extra sentence:
Hi! My name's Jan. It's nice to meet you.
Saying "It's nice to meet you" will make more of an impression than the grammar you use.
Formality depends as much on culture as it does on language. You might get a better answer if you tell us who you're introducing yourself to. Americans? Brits? Australians?
UPDATE: In the question, you asked about introducing yourself. If you tell someone "My name is Jan Wayne", they'll call you Jan. You don't need to say "My friends call me Jan". What else would they call you? "My friends call me X" is normally only used with shortened names and nicknames. You could also use it to ask someone to call you Jan instead of Ms. Wayne, but that's less common.
If your legal name is different from what people call you, then it makes sense:
My name is Elizabeth Taylor. You can call me Liz.
My name is Theodore Olson. I go by Ted.
My name is Janet Wayne. My friends call me Jan.
Sometimes people use their middle name instead of their first name:
My name is Walter Bruce Willis. You can call me Bruce.
or:
My name is Walter Willis; I go by Bruce.
But it would be weird to say:
My name is Adam Haun. I go by Adam.
