Well I'm not a 100% FAST relay racer but here are tips that could guide you to a hall of fame!

Tip One :
Click clearly and fast at the tele when the host says "R" Remember to say R too! Don't always keep your eyes at the tele , watch the speech bubbles too incase host says "GO"
Tip Two :
Ensure you have a fast room loader , if not. If you enter your room , simply click your chair quickly but clearly. After you're going to your first chair , calmly click the chair infront of the chair your heading to. That would ensure fast racing.
Tip Three :
Don't Tele Block! If you block players from racing , they might catch you up and get you in the end.
Tip Four :
Don't ever get angry at a opponent beating you , its just a game! It may result in you loosing to them or getting kicked out of the room.
Tip Five :
There's no P2Ping in these games , it might result you in getting auto kicked.

That's all for now! If I have time I might even tell you how to Win at other games!