if you want something more classic, may be a common response, but night of the living dead is definitely a good pick for something that isn't too over the top with its zombies.
if you want something more comedy leaning i'd say the zombieland movies or shaun of the dead. i had a good time watching them anyway.
if you want something more intense you could try WWZ, some people don't like it which i can understand but honestly based on intensity it's very good. definitely conveys the urgency of a zombie apocalypse. same with 28 days and 28 weeks later, bonus points for realism.
you could also try the girl with all the gifts, it too does a cordyceps based zombie apocalypse however i'm not too sure how to feel about it.
for shows, obviously the first suggestion is the walking dead however you've probably seen it, regardless it's a rec.
black summer is also decent, though some people think season 2 is confusing.
i can't remember any more lol, hope i helped somewhat!