Before we start, I want to have you to pose this question: what is linguistics? The answer is simple:
Linguistics is the science that studies language.
Simple as that. A science, just like Physics, Biology, Sociology or Chemistry.
But please don't mix up things. A linguist is not someone who study many languages. Who talks many languages is a polyglot, not a linguist. We can study some of them, but it's not the main goal. (if you want polyglots, go to this fine website)
Moreover, linguistics is not grammar. Grammar, in a narrower sense, is a traditional (and mostly unscientific) study born two thousand years ago in Greece. Linguistics works with different methods and has a much broader point of view. It's much like Alchemy versus modern Chemistry. Although Alchemy has a point about some stuff, it is not a science at all.
This thing that is commonly known in the school as the study of grammar has as its main concern the differentiation between "right" things and "wrong" things. For instance, grammar may say that "he do" doesn't exist. Well, a significant number of people say it, so it does! You may not like it, but it's there and it's part of a certain variety of language. It's pretty much like an astronomer saying that "the Moon is wrong". It can't be right or wrong, it just exists. (This subject is quite polemic and I'll talk about it again. For now, it's enough saying that it is not a blog on grammar).
Finally, "grammar", as in the scientific study of the structures of a language (without the right-or-wrong stuff), is a science and its a field of linguistics. Again, linguistics is not grammar. There are many other fields in linguistics, the grammar being just one of them.