Besides apostrophes, commas, and question marks, I have always considered special punctuation as a kind of cheat. If you're using special punctuation, there is almost always a more eloquent way of articulating your sentence/thought.
One exercise I have employed, when organizing my writing, has been to write without punctuation and in lines, every thought on a new line. Like a poem. Punctuation is essentially there to separate these thoughts, so you can start by adding the periods, then arranging your thoughts into a paragraph.
Use of special punctuation, in my opinion, is mostly a means of controlling rhythm. And the ability to ease into the rhythm of writing is largely dependent on a reader's, or writer's, familiarity with the language.
So, I would focus on periods and commas first, until you have a good familiarity with the language. That is my advice to you.