The challenge with a smaller landlocked language like Hungarian is that there's not all that many TV shows being produced and so there's less to choose from. With that being said, here are some TV shows I enjoyed:
The chronicles of the lives of young urban adults in Budapest.
Easy to watch, nice plot, fun.
People in their early twenties go on vacation on the Balaton.
A corrupt businessman and family man attempts to navigate his way through increasing domestic, legal, and financial problems.
Excellent ratings. Dialogues are hard to follow though, even with a B2-C1 level. Feels both very Hungarian and very modern. Some epic scenes and cliffhangers.
There's a lot more TV programs being produced of course, including a lot of reality show, but they're not my cup of tea.
If you like watching TV to learn languages, watching dubbed versions is kind of inevitable. Some favorites:
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Follows the personal and professional lives of six twenty to thirty-something-year-old friends living in Manhattan.
A classic. It's short, visual, uses vocabulary from daily life, and you probably know the story and many scenes by heart. Good for all levels.
A high school chemistry teacher diagnosed with inoperable lung cancer turns to manufacturing and selling methamphetamine in order to secure his family's future.
Dialogues are pretty hard to follow, but the great writing, acting and cliffhangers made me want to keep watching anyway.
A down-to-earth account of the lives of both illustrious and ordinary Romans set in the last days of the Roman Republic.
A good mix of dialogues and action. Felt easy to watch.
Something I tried is watching old favorites dubbed in Hungarian. This way I indulge in an old pleasure and it takes less effort for me to understand. Hungarians dub everything so you do have a lot of options if you go down this road.
If you live in Budapest, I'd also recommend Treadstone, a TV show based on the Jason Bourne series of books and movies. It's American, but filmed in Budapest and parts of the story take place in Budapest as well. It's fun to keep an eye out for different neighborhoods (e.g. scenes supposed to take place in New York or North Korea were actually filmed in BP!)
Credits: the descriptions for the TV shows were borrowed from IMDB.