Since some people have mentioned it already, I play as Tails in all modes except Hide and Seek, where I switch from Tails to Knuckles when I find my spot.
Co-Op: I mostly seek out the level's Armageddon Shield and use it when I want to on my way to the end of the level. I sometimes go on a ring hunt as well. I usually do not hunt down the emeralds, since I'm a Tails user, but if I end up in the Special stages a few times, I help out the Sonic players.
Match/Team Match: I mostly seek out the Grenade and Scatter Rings, as they are my most used weapons. I also grab a shield before I start hunting. When I find someone, I jump up and down while I'm moving to make myself harder to hit, while shooting at the person. I only fly away if I do not want to fight at the moment (for instance I lose my shield). If I get hit without my shield, I recover as many of the lost weapons as I can, shoot the players around me, and fly out of there. Sometimes I stick around even if I get hit, though. I do not use the mouse, mostly cause I don't move the camera to try to hit my opponents.
Race: One word is all I need to say as to why I don't play Race: Thok.
Tag/Hide and Seek: Sometimes I go after an Elemental Shield, but otherwise I mostly stay in a relatively safe area, though I sometimes move around the arena a bit. Ironically, I actually fly more often if I know I am not being chased, as jumps throw the seekers off more often than flying. For Hide and Seek, I pretty much know the areas enough to find almost everyone. I do mostly climb on the walls as Knuckles, but sometimes I'm on the ground as well, but switch to Knuckles to confuse the people hitting the TAB button to see who everyone is playing as. I tend to F12 other people when nothing much is going on, but not in Tag when the IT players are moving, and I F12 the IT player in Hide and Seek.
CTF: From the modes I play the most, I am perhaps at my most difficult in this mode. What I do varies a bit, but I'm actually a tad more interested in emeralds than the flag. I am not selfish when it comes to deciding who gets to be super. In fact, I teamsay to ask who wants to become super if I do not have all seven emeralds. I am almost always seen with a shield, because I do not want to lose what I collected. I normally pick the Force Shield, but sometimes I use a different shield. In Nimbus Ruins, I almost always have a Whirlwind Shield, to help me navigate the area. As for the flag, I have adapted a new technique of mine where I toss the flag in a "safe" place while I look around my current area to see if the other flagholder is nearby. If nothing interesting happens, I go back and grab the flag. Sometimes I do this several times, and only a couple of times has someone grabbed the flag while I was doing this. As for when I don't have the flag, I am VERY persistent in invading the bases. If I get hit, I'll grab a couple of rings and continue running towards the flag. I also have a habit of grabbing the base's flag anyway even if I get hit, and I have zero rings. Sometimes I manage to spin dash out of there with the flag. When I have the flag and I know I am being chased, I jump up and down to decrease the chances of getting hit. I sometimes take alternate routes to shake off my pursuers as well.
I do not let the fact that I pick the slowest character take advantage of me. I WILL painfully pwn Sonic players if they let their guard down.