My fondest memory of GTAIII has to be experiencing and exploring all the greatest glitches in the game, such as:
Figuring out how to get inside buildings (in particular, Joe's Garage or anywhere else that featured a unique interior) using a car and a Speed Up Gameplay code.
Looking for a so-called Purple Esperanto.
The mystery of the Seven Lifts
Finding the "You Know You Weren't Supposed To Be Able To Get Here" sign - genius.
Seeing Ray text us via beeper "See you in Miami!" and knowing that VICE CITY was going to be the next game in the series, and seriously being so excited for it.
And the best one, finding out that the game's first cutscene's location is hidden behind Shoreside Vale, dubbed "Ghost Town" - and all the Dodo flight/Tank flying that it took to get to it because it was immensely fascinating. Looking back at it, it's still so great, yet so small (compared to things like SA's hidden Liberty City and now V's North Yankton) but it started all the huge, hard-to-reach places and glitches that I love GTA for now.