Gunstar Heroes - Sidescrolling run and gun. One of my favourite games of this type ever made. Fast, frantic, and brilliant bosses, like all Treasure games. Speaking of Treasure games...
Alien Solider - Another Sidescrolling run and gun. Insanely difficult, one of the hardest games I've played on the Megadrive.
Mega Turrican - Sidescrolling run and gun with platforming elements. Reminds me a bit of Contra, Megaman, and Metroid. Good music too.
Sonic 1,2, Sonic 3 & Knuckles and Sonic CD - Platformers. Everyone knows what these are, I'm sure, but if you've never played a Sonic game, give these a spin
Streets of Rage 1, 2 and 3 - Beat 'em ups. Brilliant games, brilliant soundtracks. If you want to just play the best one, for for the second one.
Golden Axe 1, 2 and 3 - Another set of Beat 'em ups. Classic Sega gaming, although I've heard the arcade version of GA1 is better, so pick it up on MAME if you can.
Shinobi III - Hack and slash platformer, I suppose. Better than it's predecessor, The Revenge of Shinobi, but that's also a good game.
Rings Of Power - Isometric medieval fantasy RPG. A bit hard, I never got to the end, but I still found it entertaining. Sadists will like this, as you can attack anyone.
Comix Zone - Side scrolling beat 'em up set in the panels of a comic book. Very inventive as the character jumps between and rips through the panels of the comic.
Kid Chameleon - Old School Platforming, similar to Mario. Use different suits to gain different powers.
Snatcher (Mega CD) - Visual novel/point and click is the best way I can describe this I suppose. Really great and engaging plot, and half-decent voice acting.
Thunder Force III - Side scrolling shump like R-Type.
Rocket Knight Adventures - Haven't played the first one like Prometheus suggested, but this one is still a blast to play.
Wonder Boy in Monster World - Side scrolling action RPG. I really, really like this game despite it being a bit clunky.
That's all I can think of off the top of my head right now. Enjoy!
Edit:
Mickey Mouse Castle of Illusion is an absolute classic. Still good today.
Oh hell yes, I remember getting that for my birthday when I was around 10 or so. Also look out for Quack Shot starring Donald Duck and World of Illusion starring Mickey Mouse and Donald Duck. Can't believe Disney used to actually make really good games.