I wouldn't recommend writing fanfic. You're using characters already imbued with personality and style, so you aren't forced to learn to create a character on the page. Also, you can't really show fanfic to somebody you hope will hire you unless they are also a fan of the same world. Also, dialog meant to be read and dialog meant to be performed are entirely different beasts.
I would recommend writing spec screenplays. I would also recommend taking an acting class, to learn how a scene pulses between the characters, and to understand how every character needs to have an intention and a goal in every line they speak.
I would also recommend your choice of writing and performing your choice of poetry, standup comedy, or rap. I wrote poetry in high school. I still fuss over the syllables and word placement of a line the way I learned to do then. It matters which is the last word in a line of dialog. It matters when in the line the other character clocks what the character is talking about/asking for/doing.
... yes, rap. Study on how Eminem twists words around his rhymes, how he fits them into the rhythm. Great dramatic dialog has a music to it. And having to find rhymes (and half rhymes and double rhymes) that communicate what you want to say makes you realize how many different ways there are to say something, and how many you have to go through to find the best one.