I didn’t start reading webtoons until I had studied quite a bit, so I’m afraid I don’t have much to offer. 
I’ve heard Penguin Loves Mev recommended a lot (though I haven’t read it myself) - an autobiographical story about the lives of a mixed-race couple (Penguin is Korean, Mev is English). It starts out in Korean, but I think later in the series it becomes bilingual (might be when they move to the UK).
You might want to check out Webtoons 웹툰 [ where to read • recommendations • discussion ] and browse some of the sites listed (even on the paid sites the first few chapters are usually free).
It’s usually recommended to start with slice of life or romance with a modern setting as these use common, everday language, but if there’s a different genre you enjoy reading, I’d say go for it - reading something you’re interested in will be more motivational and rewarding.
If you’ve got some favourite webtoons you’ve read in translation, you could try reading the originals - knowing the story and characters already makes it so much easier to understand. I had read 비의도적 연애담 시리즈 | L28?? several times in English before attempting the Korean original, which made the experience a breeze!
Good luck! 
