Co-op: Ask for help if needed, if not, just blast on ahead.
Race: THOKTHOKTHOKTHOKTHOKTHOKTHOKTHOK. But use Mouselook for quick turning, though. It can be important. Never use the defaults either.
Tag:
When it: Use Tails, and grab as many explosion, grenade and bounce rings as possible (and rail if you're an awesome sniper). Alternatively, you can use Sonic for catching the opponent, as they can move. Catch one, and they'll start catching others, as tagging others will act like a plauge: the ones you tagged will tag others, who will tag some more, and so on.
When not it: Hide, and if found, get out of there, switch to Sonic when you can, and THOKTHOKTHOKTHOK!!! If you're in a tight spot and you've got at least one ring, thok for any damaging elements: Lasers, spikes, slime, lava... anything that damages you. As soon as you get damaged, thok away, grab a ring or more, and repeat if necessary.
HnS:
Seeker: Think. Where could a Knuckles or a Tails hide? Head for those places. Also check any silly places: Take your time, stop often, check on stupid places. Also, you can tab to get a hint about where they're hiding, that is, if they use camouflage.
Hider: Get to hard-to-find or hard-to-get places. Try to blend in with the walls as much as possible using colors, but NOT THE FLOOR. If you know the location of an elemental shield, get to it as fast as you can and go hide (avoiding any hazards that can hurt said shield) in a convenient place. There's also anti-camouflage: going in a place where your color's convenient somewhere else. This will trick taggers into thinking you're somewhere else. Avoid hiding in groups too, you'll only get hit harder.
Avoid shields if you're not hiding in a good place: they're totally visible no matter what.
Match: Grab your favourite weapons and get shooting! Strafe as constantly as you can, and don't forget to use mouselook. Grab barriers, too: you'll protect your precious chaos emeralds. If you've got everything, but can't transform thanks to a shield: get hit. Double jump instantly and get mauling. Sonic: Only thok for getaways, as some snipers are used to, well, sniping thokkers. Tails: Avoid flying near snipers. Spinflying's a good idea, however, and if you're a sniper yourself, you can play death-from-above by sniping them out in midair! Knuckles: If using OpenGL, get as high as possible, and bombard your opponent from the sky while gliding.
CTF: Same as Match, except:
Monitorbounce when you can, specially if you've got the flag or if the monitor's in the water, as you'll go back up with twice your momentum. Know any sneaky routes? Use them. If it's an underwater route, spindash through it: Water doesn't affect their speed. Take any safe routes, and be Sonic when you've got the flag. His speed will come in useful. In Silver Cascade Zone, whirlwind shields are useful: If used at exactly the right time, you can go from side A to side B in a jiffy and get the enemy's flag captured. Use rings with high knockback, too: if you're striking someone with the flag, it'll take them a bit to get back. Change routes when you can, and get to high ground if you have the flag. If you're stuck at your base and your flag's taken as well, don't camp at the beginning: move around constantly in the base. Also, traps come in useful: There are several spots where you can smack your opponent from where they don't expect it. Use teamsay when you can as well.