I read in a Japanese language learning book a sentence written as "始めまして、李です。どうぞ宜しくお願いします". I can't understand the -te inflection 始めまして, what does it represent and what meaning exactly it holds especially in this sentence or context.
It should have been 初めまして (different kanji). This shouldn't be taken as a normal te-form, it is a set phrase used when meeting people for the first time. You can take it as (roughly) "nice to meet you".
My dictionary says 初めまして is 連語. Not too sure which 2 words are they talking about. Anyway as toritoribe said it's not te form of anything. Treat as exception. The only other one similar to this one is どういたしまして when responding to apologies. That's about it.
Btw for Japanese greetings it is a good idea to treat as set phrases and not analyze the grammar. These greetings have a lot of historic baggage and can't be easily explained by modern Japanese grammar.
さようなら is actually made up of 左様 and ならば and all it means is well then if that is the case. ありがとう comes from 有り難くwhich literally means hard to come by. Best just accept these things on face value.