The Red Coins in this course are not difficult to obtain, as all of them are outside of the igloo. However, there are two of them on the frozen pond below the Chill Bully, which means you can only get them via the Koopa shell. If you happen to lose the Koopa shell before obtaining those Red Coins, then you will have to start over, as the shell doesn’t respawn. When you begin the course, move to your right and go past the tree with the warp, where you will find Red Coin #1. Ignore the coins on the frozen pond and continue past it, to the open field, where you will find Red Coin #2.
Make your way to the freezing pond now and use the Spindrift to get to the upper ledge, where you want to break open the right box to unleash the Koopa shell. Once you have it, there will be Red Coin #3 near you, so collect it and surf the upper section here for Red Coin #4, Red Coin #5 and Red Coin #6. Note that a Mr. Blizzard will pop out of the ground after the fifth coin, so be careful not to accidentally run into it and lose your shell. After collecting the last coin up here, drop off the ledge and ride on over to the two Red Coins underneath the Chill Bully. The star will spawn near your starting position.