I'll make characters of 3-4 different classes that I care about and then I'll kill several days testing abilities. I will record combat logs, reference 2005-2006 sources, and I will be a complete jerk about nit picking things that are off and posting the things I find with quick references to proof I'm drawing information from. Okay, I won't be a complete jerk but I am so, SO tired of being told by other private server coders "we can't fix that" when I'm going to be on a server where things CAN be fixed.
I have yet to have found a private server that handles the Blackout proc for Priests correctly, excluding one older server that is long gone. That sort of thing.
I'll explore zones, check out NPC's, and bug report anything that's visually fishy too with screenshots and references. Go play on Valkyrie or other servers and check out the midget sized Tauren guards in the Barrens. That's the sort of thing I'm talking about.
Somewhere down the line I'll test harder quests, especially escort quests, that are broken on other private servers and take screenshots and report on problems I find. Off the top of my head, the "Get me out of here!" goblin girl rescue in Desolace comes to mind, along with saving Ringo in Un'goro along with the robot ape, and the Night Elf in the demon cultist den in Felwood. There are tons of quests like this that are touchy, and I'm eager to kill hours looking into them. I'm weird like that.
In short, I want to test tons of shit that's always broken in other places and report it because I'd rather have a shot of doing it now as opposed to being surprised with a bug later while I level. I mean, I'm going to find bugs regardless, but what the hell. I'd have fun doing it anyway.