To be honest,i never digged into internet games at first.Back then,i was sticking to mobile because it had games which you can play anytime and anywhere,and the mobile port of my fav Sega CD game has just been ported onto mobile in 2011 aka. Sonic CD.
The first Internet games i found was Mitch & Titch trilogy.The game focuses on 2 players controlling 2 troll brothers- Mitch & Titch- as they must see their grandma,but of course,with some dangers.Mitch is green and heavy meaning that he can destroy stone blocks by simply walking into them,but he is slow.His brother Titch is blue,small and fast.Because of his smallness,he can go under few small tunnels that Mitch cant go through.
The trilogy is avaliable on Friv,and i recommend you to check it out if you like playing in 2 person modes.There is a also a option to play by yourself,but the 2 player mode is way easier.