えぇとね...mk, let's start with the place everything starts. The beginning! Here is つかさ and 瞬介's first meeting:
Shunsuke: How do you do? I'm Shunsuke.
Tsukasa: How do you do? I'm Tsukasa.
Shunsuke: It's nice to meet you.
Tsukasa: It's nice to meet you too.
(Shush, it's the first thing in ようこそ...not my fault that it's so dumb. XD)
Mk, explanation time! はじめまして and どうぞよろしく are both things said when meeting someone. hajimemashite is literally "for the first time" and douzo yoroshiku is just a blob of politeness that my book says "please regard me favorably". This is a dumb lesson, but now we have bis meeting, and that's good right? Anyways, you don't have to say both はじめまして and どうぞよろしく, one or the other is usually fine (but I guess the book wanted to cram both into one little dialogue).
です is one of the most important non-verbs ever. It's a "copula" or pretty much an "equal sign". But that comes later, with wa/ga etc. Second lesson, I promise. XD For now, just know that it's important.
The first lesson is always useless anyways. I'll right another one right now. XD
Shunsuke: Hajimemashite. Shunsuke desu.
Tsukasa: Hajimemashite. Tsukasa desu.
Shunsuke: Douzo yoroshiku.
Tsukasa: Douzo yoroshiku.