Puyo Pop shares the ruleset that Mean Bean Machine uses and is kind of a "reboot" of the original Puyo game, wouldn't particularly recommend it
Puyo Pop Fever is focused around the eponymous Fever game mode, which is more beginner-friendly but still has some competitive appeal. It still has the classic modes available in free play, of course.
Compared to the GBA version of Fever, the DS version is basically a straight port that takes advantage of the new hardware. So you gain wireless multiplayer, single-cart 8 player multi (up from 4), and dumb but thankfully purely aesthetic dual screen / touch / mic gimmicks.
I do think 20th is the gold standard of Puyo games and beats out Fever in pretty much every respect (asides from having no English translation), especially since it has structured lessons and drills for beginners unlike Fever. We cover the same content (without the hands-on tutorials, obviously)
here. Try out Fever GBA/DS (these were the games I started out with) and if you want to gather more Puyo games from there, pick up 20th.