An introduction usually has three parts.

1. はじめまして which means something like "we're meeting for the first time"

2. You say your name (and probably wait for the other person to say their name), in this case 田中です

3. 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.

the subject has to come first. i believe.

Thats what I said

in your case, i am pretty sure 田中 would be assumed as the other person in the conversation instead of the person who is actually speaking. The answer being 田中ですよろしく。

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.

if you open DuoLingo on web browser, there's a review page that explains the context of each lesson :)

it's beside the key icon before you start the lesson

はじめまして、田中です, should probably be accepted, even if they’re trying to teach the “よろしく” concept. But even if you said はじめまして and then your name, both parties would inevitably follow it up with “よろしくおねがいします。”

はじめまして is more of a formal greating. よろしく is you as a polite greating after introducing yourself.

I always use どうぞよろしくお願いします (douzo yoroshiku onegaishimasu) when introducing myself, but I only use はじめまして (hajimemashite) if the other person does. They rarely do, so I've almost stopped using はじめまして completely.

田中です、よろしくお願いします was accepted for me, I think if you submitted an error report the hiragana version will also be accepted.

You want to say はじめまして (hajimemashite). Yes, that's the order that most people use. Hajimemashite usually comes first, then your name, then yoroshiku onegaishimasu.

You dont put san in front of you're own name.

The only scenario i can think of is when you are practicing a play or practicing foreign language in class, and you're like, am i/ I'll be 田中さん。

I don’t think people usually refer to themselves with さん、not that it’s wrong. Otherwise I don’t see anything.

It's like talking about yourself in the third person. It's frowned upon in English too.

よろしくお願いします (yoroshiku onegaishimasu) - formal, polite

どうぞよろしく (douzo yoroshiku) - more casual

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.

I think I've managed to figure it out - I had a space between "Tanaka desu" and "yoroshiku". Spaces are a bit of a no-no in Japanese sentences, it seems.

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:

田中です、よろしく

田中です、よろしくおねがいします

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)

I'm Tanaka.

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.

どうぞよろしく with no space between the words should be correct.

Read elsewhere in these comments. Querig explains the literal differences. English speakers just do not usually say all these extra courtesies, "nice to meet you" is easiest to explain to beginners.