Amiga provided a lot of first time experiences.
The first Amiga game I saw running was Battle Squadron, a scrolling shooter which at the time was arcade machine quality, with amazing music, just blew me away.
Stunt Car Racer was probalby one of the first "LAN party" games I ever played. Serial Network IIRC and it was awesome.
MusicX, first MIDI composing experience, linked to a rubbish Yamaha PSR keyboard. First time "tracking" music with Protracker, MED and then OctaMED.
Bomberman, first time I played a game using 2 joysticks! We used to play 4 player bomberman with 2 people and it was just utter mayhem quality fun.
Commodore RGB monitor, first colour (RGB) monitor I ever owned.
I'm sure there are many other firsts
Nobody mentioned Shadow of Beast yet? The first game made a software publisher (Psygnosis) from pure hype?
On a related note, games-wise History Line was also superb with 2 players and hasn't been metioned. (The Settlers post reminded me of it). Flood was probably the first Bullfrog game I ever played.