I'm used to considering myself a veteren because I started playing during Time Spiral Block. You guys started way back in the day! I was only a little baby! (I mean that literally, by the way. I'm 24 as of 2 days ago)
A brief history of Magic by set (blatantly stolen from Wikipedia). Core sets are italicized, Blocks alternate being bold (i.e. one block is bold, next block is not, etc): For most of Magic history, Blocks followed a 3 set structure of large set/ small set/ small set, although they start playing with that from Lorwyn onwards.
1993: Alpha, Beta, Unlimited, Arabian Nights
1994: Revised, Antiquities, Legends, The Dark, Fallen Empires
1995: Fourth Edition, Ice Age, Homelands
1996: Alliances, Mirage
1997: Visions, Fifth Edition, Weatherlight, Tempest
1998: Stronghold, Exodus, Urza's Saga
1999: Urza's Legacy, Sixth Edition (major rules restructuring), Urza's Destiny, Mercadian Masques
2000: Nemesis, Prophecy, Invasion
2001: Planeshift, Seventh Edition, Apocalypse, Odyssey
2002: Torment, Judgement, Onslaught,
2003: Legions, Scourge, Eighth Edition (new card frames), Mirrodin
2004: Darksteel, Fifth Dawn, Champions of Kamigawa
2005: Betrayers of Kamigawa, Ninth Edition, Saviors of Kamigawa, Ravnica
2006: Guildpact, Dissension, Coldsnap (Coldsnap was nominally part of the Ice Age block, although Ice Age predated blocks), Time Spiral
2007: Planar Chaos, Future Sight (major storyline restructuring), Tenth Edition, Lorwyn (Introduced Planeswalkers as cards)
2008: Morningtide, Shadowmoor, Eventide (Lorwyn/Shadowmoor was a split block, in the format large/small/large/small, Shards of Alara
2009: Conflux, Alara Reborn, Magic 2010 (rules update, core set restructure), Zendikar
2010: Worldwake, Rise of the Eldrazi , Magic 2011, Scars of Mirrodin
2011: Mirrodin Besieged, New Phyrexia, Magic 2012, Innistrad
2012: Dark Ascension, Acacyn Restored, Magic 2013, Return to Ravnica
2013: Gatecrash, Dragon's Maze, Magic 2014, Theros
2014: Born of the Gods, Journey into Nyx, Magic 2015, Khans of Tarkir
2015: Fate Reforged, Dragons of Tarkir, Magic Origins (last core set, end of the three block structure), Battle for Zendicar
2016: Oath of the Gatewatch, Shadows over Innistrad, Eldritch Moon, Kaladesh
2017: Aether Revolt, Amonkhet, (coming soon: Hour of Devastation, Ixalan)
There are also a variety of special sets (Unglued and Unhinged, Modern Masters x3, and Conspiracy x2, but you can look them up yourself.
I joined Magic half way through Time Spiral block (which was a very strange place to start. Imagine you're a middleschooler who's never played before, and someone hands you Argent Adept and tells you to go have fun. I loved it.) I was never really a serious player. Nowdays I show up for about half the prereleases, and that's good enough for me.