First of all, I typed the ship in search column, then I picked the completed stories, after that I chose the genre. In the end, I would like to vote up the story when I like the story. However, I would like to leave in the middle of story when I felt bored.

The storyline is the most important thing, but I rarely ever read stories about "couples" which I don't ship at all.

i've never thought about this until now. but i guess it'd be the ship and the storyline together?
well for me, i'd browse the tag of my ship first, then search if there's any story interesting enough to read.

Everything, especially the writing and the storyline.

My first priority is the artist, then the story. For me the ship is at the last

All of them together are very important to me. if one of them is lacking then the story is not worth it.

to me its the writing
if one is a good writer, they can make even a super cliche fic enjoyable

storyline, characters, then ship for me.

I'm going to say ship because that's what got me into writing fanfics.
But yeah, I do agree with what just1unnie said: storyline, then ship.