I tend to like the very sad and intense so am leaving out a lot of the dramas I'd normally recommend to jdrama beginners, but a few that I enjoyed that you might be interested in are:
Nobuta wo Produce: school drama about two classmates who give a makeover to an unpopular student and find themselves in the process. Gets a lot deeper than it sounds.
Hanzawa Naoki: business drama about an underdog businessman rising up to the occasion, the highest rated Japanese drama by amount of watchers in Japan in recent years. It sounds dry from the description but is really great, especially if you have experience working for a boss or company with unfair and hierarchical power structures.
Mahoro Ekimae Bangaichi: slice of life drama and episodic, but really sucks you in. Fantastic characters and acting.
Love Shuffle: I didn't like this one much but most everyone does, even those not really into jdramas. A romcom by a writer I usually really love, that goes in a different direction than most romcoms. A group of people decide to swap partners to see if they are really with the one they were meant to be with.
If you are willing to do a little more intense: The Quiz Show and The Quiz Show 2 are great, and similar to the Korean version of Liar Game.
If you are up for movies too, Initiation Love is fantastic!! I fell absolutely in love with it and it has a great surprise ending. I highly recommend it.