Definitely not one of my best runs on this level, but it will do. Also, have you guys noticed that every level starts with the letter "R?" I just noticed that, probably because it's named Sonic R. There's also two Chaos Emeralds in this stage.
This is the Nintendo Gamecube version of Sonic R played on Sonic Gems Collection.
Radical City is a pretty easy level as well.
Coin #1: 0:31, as you see, there are two paths to go in the beginning. In the beginning of the left path, you should see a little hole. Drop down into there to get the 1st coin.
Coin #2: 1:03, as you go a little bit farther in Radical City, you should see a little sign that's black and yellow. Pass through that and you should see the second coin.
Coin #3: 1:13 After jumping out of the pinball area, open the door that cost 20 rings to run through the 3rd coin.
Coin #4: 1:32 From where you got the 1st Chaos Emerald, turn left, jump over the water to land near the 4th coin
Coin #5: Go up the road from where you got the 4th coin, and you should see another black and yellow sign. Pass it, jump and collect the last coin.
Chaos Emerald #2: From where you got the first coin, pass through the door and collect the second Chaos Emerald.
Chaos Emerald #3: Pass through the black and yellow sign that led you to the second coin and go through the door that cost 50 rings. Run all the way and get the third Chaos Emerald.
The character that you will face to unlock is the, Tails Doll. The Tails Doll isn't a bad character as well, but is a bit slow. He can stay above water as long as you want, though.
Enjoy, and check out for the next stage, Regal Ruin.