As a Torah reader for about 25 years, my advice may sound somewhat unconventional, so accept this as my own method that worked for me. It may or may not work for you.
First, learning Hebrew vocabulary and grammar is certainly a good and important start. But, you should also learn Biblical Hebrew, as its grammar is markedly different from "Modern" Hebrew. Most notably, the Torah uses "Vav Hahipuch" which is a method of adding a vav to the beginning of a verb that "flips" a future verb to past and past tense to future. This is one of a few examples where Biblical grammar may be different from modern Hebrew.
It is not necessary to know the meanings of every word in the Torah, but a fairly strong command of grammatical rules will help tremendously in Torah reading. It's not necessary, but it is a big help. It's not 100% required as you could read a Torah portion without knowing the meaning of any words - just by memorizing what to read. But, if you know the meaning, it will help with the verse's grammar. The trope (notes) is primarily for grammatical purposes - not just to sing them.
(When I read, I'm sometimes unsure of the trope. But, since I understand, mostly, what I'm reading, I can tell that if I group certain words with the wrong trope, the sentence meaning may change, or it won't make sense at all. Certain notes have pauses, similar to a comma in a sentence. If you pause (comma) in the wrong place, you change the sentence's meaning or it doesn't make sense.)
It is essential that you practice recognizing Torah script - the shape of the letters in the Torah is significantly different from standard Hebrew font. This may sound trivial, but in my shul, I have a majority of people that have no problem at all reading from a siddur or Chumash, but only about 5 people can read Torah lettering. Along with that, it also helps to have a good vocabulary of most common Hebrew words and know how to read them without vowels. This second skill is not 100% necessary for Torah reading, but it will ease the challenge.
Next, is trope. I've seen almost every Bar Mitzvah teacher teach the names of the trope and how they sound, prior to combining the trope with the actual Torah words. Unless the Bar Mitzvah boy will become a continuous Torah reader, this is a waste of time, IMO. In your case, however, this skill is valuable, but also not 100% necessary.
OK, HOW do you learn trope?
I learned in a pre-computer era. I sat in shul and listened carefully to the Torah reader and followed the notes in the Chumash with listening carefully to the reader. I would focus on a few notes each week until I felt comfortable, then I would focus on another group of notes.
Eventually, when I got all the notes and sounds down, I used a Tikkun - which is a book used for studying the Torah reading. Each page has 2 columns - the right side having the font and verses as they appear in the Chumash (in "regular" Hebrew) and the left column with how it looks in the Torah scroll. From there, it's memorizing. Do a few verses at a time - start with the normal type column and "test" yourself reading from the Torah column. Of course, the more often you practice, the easier it becomes. Keep in mind, that after all these years, I have pretty much memorized the entire Torah.
There are, now, several good computer Torah reading programs available. One of the best ones is called Trope Trainer. I use it, mainly, to print the weekly parsha sheets and I study the parsha during my daily commute. However, I think it is a great disk for a new learner. You can adjust the speed and pitch "level" (tenor / baritone / bass, etc.) to your liking. There are even several "accent" options (Israeli , various Hassidic styles, etc.) I also like that it has an option that lets you click just one word at a time so you can hear it sung with the note, or go a whole verse or an entire section at a time. This is excellent for reviewing. The disc also has separate tutorials that practice just notes and note "groups" (certain notes commonly appear together).
Supp: See @Shokhet's comment, below this answer. I should have researched this linked answer, myself (too much pre-Shabbat soup made me lazy :-) There's much good practical advice there!
I know I gave a lengthy answer, and, again, my method has several ideas that, hopefully, will work for you. If you can find a Torah reader or some other similar mentor to help you, that would be wonderful. And, remember that the mentor can be online as well. Skype works quite well!
I commend your efforts and wish you success. The population of good precise professional Torah readers, I think, is diminishing these days.