"Nice to meet you, I'm Tanaka."
No, when you are referring to yourself -san is not used. It is just used to politely refer to other people but never to yourself. When you say your name and want to talk politely just add -desu, and if you want to be informal just drop -desu, although it can be considered very rude to avoid the use of -desu.
I think you might be misunderstanding a little (which is completely understandable since many comments try to simplify things as much as possible).
Functionally, 「田中です」and「田中といいます」are both statements that can be translated to "I am Tanaka." Implicit in both of these is the subject, usually assumed to be the speaker.
However, this is where grammatical differences come in. In the case of「田中です」, explicitly including a subject, e.g. かれ(he/him) or かのじょ(she/her), simply changes the translation to "he/she is Tanaka."
For「田中といいます」, it's not so simple. いいます can also be written as 言います, which means "to say". Basically, what is implicit in this case is something along the lines of "Tanaka is what I say about myself", which often becomes translated to "please call me Tanaka (because that is what I call myself)". It's considered more polite than です because it's very deferential, like you're saying "I call myself Tanaka, but you can call me as you like" whereas です is kind of saying "this is what I am (like it or not)." The tricky thing about といいます is that it's exactly the way you would quote what someone else says. Therefore, if you change the subject (in order to introduce someone else), the sentence changes to "he/she says "Tanaka" ", but no longer functions as an introduction, rather an introduction of the strange things he/she likes to say. That's why it is only ever used to introduce yourself.
To explicitly ask someone to call you by a certain name, you need to use a slightly more advanced grammar structure. For example, my name is Joshua, but I prefer people to call me Josh. So when I introduce myself, I would say「ジョシュアといいます。でも(But)ジョシュと呼んでください(yonde kudasai call please)」
Sorry for such a long comment, but it's hard to concisely explain all the nuance using just text.
Hajimemashite is the "te" form of the verb "hajimeru" or "hajimemasu". The verb means to start, to begin. As the "te" from may be thought of as the imperative of the verb, hajimemashite may be literally translated as "do begin!", or "let's start (our relations)!" and then "nice to meet you" for the first time you meet someone. As for yoroshiku, it may be translated as the adverb of the adjetive yoroshii or yoi, a form of "ii", that means good, well. So yoroshiku may be understood as "do me well", "treat me fine (as I will treat you)". Usually comes followed by onegaishimasu, meaning "I beg you", or just "please".
はじめまして = a set phrase/greeting used when meeting someone for the first time; no direct English translation, but usually translated as "nice to meet you"
田中 (たなか) = a common Japanese surname (family name); Japanese people are usually only called by their first/given names by family and close friends
です = the copula/verb "to be", usually being translated as "is/am"; it equates the object (in this case, "Tanaka") with the subject (in this case, the implied "I")
A little IPA under the answers dictionary style (after getting them right) would go a long way- knowing what the words look like is one thing, but being able to say them well would be great to. This goes for languages with Roman alphabets too (Irish, for example, could use it).
I know Japanese hiragana is phonetic, but its not quite- it sounds like 'u' is largely unexpressed or at least unstressed at the end of words, for example.