Getting Sonic's A ranks has more to do what you do and what you don't do, and if you do it in the right place. There are secret bonuses scattered throughout all the levels.
For City Escape, jump on all the ramps but the smaller one in hard mode. Try to get a least one "Awesome". After the boarding sequence, home-attack the Hunter, jump dash on the rail and home attack the second Hunter without touching the ground. On the next two rails, grind and crouch on both of them in a sequence without touching the ground (use bounce attack to quicky hit the rails). Then jump dash on the rail by the pit. Then, perform tricks on all the ramps without stopping to uncover a "Tight" bonus. Later on in the stage don't forget to hit the Gold Beetle. Before the loop, make your way to the top of the bridge, light dash on the rail of rings and, after falling, on the rail of rings in the ground. Do it without touching the ground to get an "Awesome" bonus. After you overcome the big death pit, get ready for some time attack. Hit the rail with bounce attack and get as quickly as you can to the bottom. If you manage to do it within a time limit, you'll get a "Tight" bonus. Manage to hit the bar and swing yourself to the top of the staircase (still within the time limit) and you'll get an "Awesome". Finally, start the truck escape with a spin dash, to quickly gain speed. Hit jumps on both ramps and proudly make your way to the Goal Ring. Hope you got an A rank.
Metal Harbor has only 174 rings IIRC, and if you know your Sonic Adventure 2, you'll know that if you get them all, you get a Perfect at the end of the level and, automatically, an 'A' rank. Hunt for the rings first in normal, then after you know where they all are, go for hard.
Pyramid Cave - can't tell you much more than CZ64 already did. Do focus on the hoops in the long hallways within the stage - each gives you a bonus. But, if you miss one, too bad. Keep moving.
Final Rush. For time limit, pray to God. Seriously, it's probably the hardest mission in the game. It's easy to score an A, and easy to get to the goal within the time limit. Try to do both. Mission impossible.
Crazy Gadget: yeah. If after grinding or handbar sliding an enemy appears in front of you, it's most invariably a bonus maneuver. Follow the rail/bar until the end, jump off and quickly kill the beast. Choose your best attack - somersault for Shield Hunters, homing attack for Chaos P-2 and P-3 and maybe magic hands for Chaos P-1 (dividing enemy).
BTW, while we're on the subject: where is that last ring in GHZ? (Try to get them all, you'll see what I mean.)