First game was Starglider, which I had seen at a computer show not long before getting my ST. It was almost the first thing I tried out of the box. The mouse control was a totally new thing to me, and a bit of a rude awakening at first
Next thing were the Silica pack-in games like Megaroids and Time Bandit, then after that, Leaderboard, the other commercial game I ordered with my new machine, based on previous experiences on a friends C64 version.
Graphically, Leaderboard lived up to its 16-bit promise, but I quickly discovered how annoying the underside joystick/mouse ports were, as this game featured a dongle that plugged in there! This tended to dampen the appeal of that game, which didn't get too much airtime, once I started to get other stuff.