Hi Dom and welcome. I am myself new to the forum as well.
I'm not going to be really original and re-iterate what has been said so far but great starting place for games would be Sonic Adventure and Crazy Taxi series. These are both instant fun games.
If you want a fighting game with neat visuals, Soul Calibur is a great candidate. Puzzle game wise I'd recommend classics such as Bubble Bobble 4, Puyo Puyo Fever or Sega Swirl. If you are looking for a more original puzzle game I would recommend Wetrix.
I agree with Bob Dobbs, Shenmue would be better off played on a PS4, if not exclusively for the long loading times. However I understand you just got a Dreamcast and would like to use it. Either way, the game offers a great adventure.