This is an expansion of my comment above.
The first step to literacy is reading. There are stylistic conventions for written mathematics, and they are not covered in any course, though there are some prescriptive books around. As you mention, there is more than one style, though in any given branch of mathematics, most of the papers probably form a homogeneous group.
The second step is writing. Writing by itself is immensely helpful, but it's even better if a "dialect expert" comments and corrects your output. This is especially important for non-native speakers of the target language (usually English). As for the comments, don't take them too seriously. There's a difference between conventions and style, and even regarding the former, sometimes it's good to be innovative.
Finally, one way to find enough material to write about (unless you're a phenomenal researcher even at this early stage) is to write expositions of proofs (or subjects) you like. There is often more than one way to explain a proof, and you can practice both your understanding of the material, your explanatory skills, and your literacy. You can complement it with lecturing on the material.
My advice would be:
$\bullet $ Do many calculations
$\bullet \bullet$ Ask yourself concrete questions whose answer is a number.
$\bullet \bullet \bullet$ Learn a reasonable number of formulas by heart. (Yes, I know this is not fashionable advice!)
$\bullet \bullet \bullet \bullet$ Beware the illusion that nice general theorems are the ultimate goal in your subject.
I have answered many questions tagged algebraic geometry on this site and I was struck by the contrast between the excellent quality of the beginners in that field and the nature of their questions: they would know and really understand abstract results (like, say, the equivalence between the category of commutative rings and that of affine schemes) but would have difficulties answering more down-to-earth questions like: "how many lines cut four skew lines in three-dimensional projective space ?" or "give an example of a curve of genus $17$".
In summary the point of view of some quantum physicists toward the philosophy of their subject
Shut up and calculate ! contains more than a grain of truth for mathematicians too (although it could be formulated more gently...)
Nota Bene
The above exhortation is probably due to David Mermin, although it is generally misattributed to Richard Feynman.
Edit
Since @Mark Fantini asks for more advice in his comment below, here are some more (maybe too personal!) thoughts:
$\bigstar$ Learn mathematics pen in hand but after that go for a stroll and think about what you have just learned. This helps classifying new material in the brain, just as sleep is well known to do.
$\bigstar \bigstar$ Go to a tea-room with a mathematician friend and scribble mathematics for a few hours in a relaxed atmosphere.
I am very lucky to have had such a friend since he and I were beginners and we have been working together in public places ( also in our shared office, of course) ever since.
$\bigstar \bigstar \bigstar$ If you don't understand something, teach it!
I had wanted to learn scheme theory for quite a time but I backed down because I feared the subject.
One semester I agreed to teach it to graduate students and since I had burned my vessels I really had to learn the subject in detail and invent simple examples to see what was going on.
My students did not realize that I was only one or two courses ahead of them and my teaching was maybe better in that the material taught was as new and difficult for me as it was for them.
$\bigstar \bigstar \bigstar \bigstar$ Last not least: use this site!
Not everybody has a teaching position, but all of us can answer here.
I find using this site and MathOverflow the most efficient way of learning or reviewing mathematics . The problems posed are often quite ingenious, incredibly varied and the best source for questions necessitating explicit calculations (see points $\bullet$ and $\bullet \bullet$ above).
New Edit (December 9th)
Here are a few questions posted in the last 12 days which I find are in the spirit of what I recommend in my post: a), b), c), d), e), f), g), h).
Newer Edit(December 17th)
Here is a fantastic question, brilliantly illustrating how to aggressively tackle mathematics, asked a few hours ago by Clara: very concrete, low-tech and naïve but quite disconcerting.
This question also seems to me absolutely original : I challenge everybody to find it in any book or any on-line document !
Best Answer
Wikipedia has a list of 21st-century mathematicians while the book Mathematicians : an outer view of the inner world has the following list in portrait form with their autobiography: