Match:
Usually, I'll go for the Rail ASAP, then find a spot where I have easy access to rings, am protected on at least one side, is high up (for better snipage), and maybe has an item box around (for shields). Jade Valley is good for this because of that little nook in the wall near the main waterfall that contains an item box, and has a row of rings and a ring box right in front of it.
If I have no access to Rail easily, or there is no safe sniping spot (like in Meadow Match) I'll go with Bomb and just fly around throwing explosives everywhere.
Team Match:
Pretty much the same thing goes for this, though occasionally I'll follow along with my teammates and give them support when needed. Most of the time, though, I'm a lone Wolf, because most of the time my teammates run around doing their own thing.
CTF:
Same thing with Rail on most maps. If I have no easy access to Rail and have no good sniping spots, I'll still stay at base and defend the flag, occasionally attempting to run the flag if the teams are low and if I can quickly go from one base to another (Cloud Palace, Icicle Falls, and occasionally Twisted Terminal). If the map is Nimbus Ruins, I'll still grab the Rail, but I'll always try and get the enemy's flag. In Lime Forest, I'll go for the Scatter and protect my flag, always. However, lately I've noticed that Scatter isn't always registering when it hits, so I'm going back to just using Red Ring for defense in LF.
Tag: Run like Mystic is chasing me with a banhammer (again)
H&S: Find a spot, change colour, switch to Knuckles (maybe), hide on a wall. Alternately, just find a spot I can hide in that is out of the way, at least moderately hard to notice, and that I can easily change my character colour to match with at a distance. I've stopped trying to hide in waterfalls, however, because that never seems to work.
Race: Leave game because Race is still utterly broken.
Co-op: Get to the end, try to help my teammates past obstacles, and always, ALWAYS go for the Emerald Tokens. If there's one thing I never stand for in co-op, it's not getting the Chaos Emeralds.
Wait, I just realised, this thread gives people an advantage over the rest of us because now they can know our strategies. Crap.
EDIT:
Tag: When I'm hiding, I'm searching for a good place to hide. If I get noticed, I run away as soon as possible. When I'm IT, I tend to try and rail/snipe everyone.
Hide and Seek: Never played it. Don't care about it too.
... You DO realise that by hiding in Tag, you're pretty much playing H&S? The only real differences between H&S and Tag are:
a. After a time limit for hiding, you can't move
b. You can't drown when hiding
c. Only one person is ever IT
I REALLY suggest trying out H&S. It's one of the most fun game modes to play after 10+ rounds of Match/Team Match/CTF/Tag.