Bogyó és Babóca - this is for children around 3. it may contain many words you would never use in your whole life. it is contemporary.
Kérem a következőt - it is full of word-wits which can be hard to understand for a foreigner but the jokes are brilliant and the story is enjoyable for an adult as well. It is from the 70s
A nagy ho-ho-horgász - from the '80s
Macskafogó - again, from the 80s. This is a single movie, not a series.
Süsü a sárkány. It is a puppet film, not a cartoon, but it is brilliant. It is from the 70s-80s of course :)
I can also advise "Mézga Család" as well.
I am sure that many of these are available on YouTube.