[quote name='Saucy Jack']The 3 stories are linked by different characters who appear in more than one of the stories. For example, there are characters from GTA 3 who appear in Vice City, characters from Vice that appear in SA, characters from GTA 3 who appear in SA... etc.[/QUOTE]
Well, there are some cameos, but that's not what connects the three PS2-era GTA games.
The connection should be obvious to anyone who ever played GTA1 (most GTA fans came in at GTA3, and never played the first two).
The connection between them is that the GTAIII/ViceCity/SanAndreas Trilogy is, in a way, a giant remake/reenvisioning of the original GTA. In the original GTA, there are three cities: Liberty City, Vice City, and San Andreas. Each of these cities was spun off into it's own game on PS2. So, there you go. That's the connection, that relates the three games.
Anyway, my favorite GTAs are GTA2 and Vice City. But I don't really care for the series too much. It's mind-bogglingly overrated.