"I'm Tanaka, nice to meet you!"
An introduction usually has three parts.
はじめまして which means something like "we're meeting for the first time"
You say your name (and probably wait for the other person to say their name), in this case 田中です
よろしくおねがいします. This ends the introduction and in a way establishes a relationship. You now - formally at least - know each other. (It can be used in other situations as well, meaning anything from "let's have a good relationship", "please take care of what I have just entrusted to you", to "I'm counting on you" etc. depending on the situation.)
These three phrases are usually said in that order. Also, they are each a sentence of their own, which is why in the solution there is a comma. You could put a fullstop between them too, but grammatically you could not write the phrases one after the next without punctuation.
Another thing about よろしく: In this scenario I assume that Tanaka is the teacher greeting his students. In that case it's okay for him to use the short version of よろしく. As a student you would be, hierarchically speaking, on a lower level than your teacher. As such, when greeting your teacher you would definitely use the longer version よろしくおねがいします. Using the short version does not look good when you talk to someone "above" you.
Japanese is more (S)OV than SOV. The subject is frequently dropped or replaced by a topic, instead of being directly stated, like in English.
"Pizza eat" or "As for me, pizza eat" rather than "I pizza eat"
The subject is implied based on context, so "I eat pizza" or "You eat pizza" or "They eat pizza" would be essentially the same sentence in spoken Japanese.
Duolingo is more learning by osmosis than learning by explanation. When learning new alphabets and syllabaries, I think this method just does not work for some people. If you're feeling frustrated and finding it hard to progress, I recommend trying different websites to learn about hiragana, katakana, and kanji. Duolingo can be a great place to learn and study, but sometimes you need supplements to suit your own learning style.
I dont understand why theyre teaching us yoroshiku first. When meeting someone for the first time, the introducing person says hajimemashite, to which the other person replies yoroshiku. Its like a call and response format for meeting people, especially formally. Ive learned that in every other class ive taken.
どうぞ makes it a little more polite. どうぞよろしくお願いします (douzo yoroshiku onegaishimasu) would be quite polite, while よろしく (yoroshiku) by itself is casual, and どうぞよろしく would be a slightly more polite than that. If you submitted an error report, I think your answer should be added to the database. When there are so many ways to say the same thing, venturing away from the duolingo-expected answer always puts you at risk of being marked "wrong" even if your answer is correct.
If you have grounds to think your answer should be accepted as correct, you should be using the 'report' feature to say so. Many questions here do accept multiple correct answers, but I think it usually requires someone pointing out the extra answers for that to happen. I suspect questions are initially just entered en masse with single answers and then the extras are added organically as they're found.
Edit: I tested it, and as of today, it does accept both of these, at least:
When I took a Japanese class in 2004, taught by a lady who was born and raised in Japan, she stated that saying DOUZOU YOROSHIKU is the most polite way to speak to someone when introducing yourself. For some reason, when I hit the bubbles "Tanaka desu, douzou yoroshiku" it did not accept my answer.
You're conveying the same idea and you can always try submitting an error report, but I think that the contributors would want you to show that you know the difference between "my name is ~" and "I'm ~".
私の名前は田中です。 (watashi no namae wa tanaka desu)
My name is Tanaka.
田中です。 (tanaka desu)
Not necessarily. It's common in business to meet many people during the span of your career, which in Japan may be a very long career for the same company, so you will frequently meet people for a second time and yet not necessarily expect them to recall your name without prompting.
I always use よろしくお願いします (yoroshiku onegaishimasu) when introducing myself for the first time. I used to always start my introductions with はじめまして (hajimemashite), say my name, and then say よろしくお願いします, but I've stopped saying はじめまして at all after I noticed that Japanese people where I live seem to rarely use it.
Japanese rarely uses subject pronouns like "I". Have you read the tips and notes for Intro 2?
One thing that makes Japanese very different from English is the Japanese tendency to drop the subject of the sentence when the meaning is clear from context. Statements usually refer to oneself, while questions usually address the person you're speaking with.
The subject pronoun isn't included, but most people would assume you're saying （わたしは）田中です - (watashi wa) tanaka desu, I am Tanaka. If the omitted pronoun was （あなたは）田中です - (anata wa) tanaka desu, you wouldn't be addressing the person, you would be saying you are Tanaka. If you wanted to address a person, you wouldn't include です, and you would probably attach an honorific like さん to their name.
Yes, I think it would be better to use よろしくおねがいします with といいます. よろしく is rather casual, and doesn't really fit with the polite forms that Duolingo teaches. If either of you submitted an error report saying "my answer should be accept", I think there's a chance that it would be added to the possible answers, though.