Personally, I have a long Plan To Watch list that I never look through whenever searching for a new drama lol. The reason my Plan To Watch list is so long is because I find new dramas that I don't have the time to watch. Since I'm already watching several dramas at once I decided to just throw them into this never ending list. I then become overwhelmed with how long the list has gotten and don't check it when looking for a new show (yet I continue to add to the list). When I finally do find something to watch, some things I put into consideration are: the cast, description, genre, release date, and ratings.
Most of the time, when I discover dramas, it's by popularity, me looking at dramas a certain actor/actress has been in, or recommendations.
I also can't help but notice popular shows that are airing since they always seem to be featured on streaming services such as Netflix, Viki and many others. I always tap on them and read their description, then if I like the sound of it I'll begin to watch it. The thing I don't like about this method though, is if I get really invested in the show and I need to wait a week for each episode to air lol. I sound really impatient but when I start a really good drama I can't seem to stop, and start binging.
An example of these situations would be Moment of Eighteen. I'm absolutely in love with this drama! I noticed it because it was airing and was on the featured page of Viki. However, I also tapped on this drama when it popped up on my screen was because I recognized the female actress (Kim Hyang Gi) from another drama I had recently watched (Sweet Revenge). I watched Sweet Revenge because Cha Eun Woo was in it. I then started looking up things about the cast of Moment of Eighteen and found out that the male lead (Ong Seong Woo) was in a kpop group called Wanna One whom I have heard of before. I started watching tons of videos with him in it and began stanning lol. A few months ago I watched Produce 48 which created iZone, and since Wanna One came from Produce 101 Season 2 I began watching that as well.
So it kind of goes in a cycle for me, I feel like this wasn't really that helpful, just mostly me talking about useless shows and actors lol. But if you haven't started watching Moment of Eighteen yet, I suggest you should lol.