Text messing

Text messaging is a common form of communication nowadays because young people usually use this way, which has so many abbreviations. [This is OK but I am not 100% sure about the meaning.  Do you mean that young people use texts so much because they can use abbreviations?]
I'm going to compare the advantages and disadvantages of text messaging.

On the one hand, text messaging is a good form of communication for young people since most of teenagers understand this type of language; therefore, it is more comfortable and faster for them.
Secondly, text messaging is very positive but you have to use it in an appropiate situation because this then allows us to / means that we can save a lot of time. [I didn't understand your logic but the grammar was fine!] 

On the other hand, old people often don't understand text-speak, so it can be quite difficult for them to use this means of communication.
Text messaging is also negative / also has down-sides for students because they get used to writing badly / lazily / sloppily.

In conclusion, text messaging is a kind of communication which is easier to understand for young people than old people and it is a great way to communicate us if we use it in suitable adequate situations.

All good!  Texting is a "means of communication" - a good collocation.  

Often "adecuado" is translated as suitable - it's a false friend. Check the meaning of "adequate".

You can talk about "up-sides" and "down-sides" of things instead of using the words "positive" and negative" - it's just an alternative.  

I changed some of the punctuation.  You don't need a comma before "because". 

Kind regards, Michael