In regards to the Mario series you've got to at least play the original. Without that game you can't really get a good sense for how far the series has come over the years. As far as the notable games in the series go these are the ones that absolutely must be played:

Super Mario Brothers 3 (NES)
Super Mario World (SNES)
Super Mario 64 (N64)

SMB3 added several of the things that would eventually become trademarks of the series. It's surprisingly long for an NES game if you're going for 100% completion (which in this case is clearing every level on a single playthrough). SMW is even longer with 96 levels if my memory is correct. As for Mario 64 I strongly believe to this day that it's the game all 3D platformers are compared to.

For the games past 64 you'll need to ask others since the last Mario game I played was Sunshine which I myself didn't care for.