Enter Mission mode and select Mission 2, "Collect 'X' Rings" on any of the tracks. During this mission you are not limited to a set number of rings that you can collect. Collect as many rings as possible before reaching the goal, then wait for the results screen to appear. Once the results finish displaying, all the rings you have collected will be automatically sent to your ring account (although the game will not show you this). Pause game play and select "Restart" to begin the mission again. Keep repeating the process. If you can get at least 250 rings per try, you should have collected enough rings to purchase even the Rainbow gear within an hour or two.
The Gambler gear at the shop has an interesting secret. If you have bought it and take it on a World Grand Prix, you can earn 500 rings per race. If you finish in first place in the Grand Prix on top of that, you will get a total of 3,500 rings. Just three Grand Prixes will give you enough rings to buy the rainbow in exactly one hour. Note: You must know your courses well though, because the Gambler cannot take type shortcuts.
Enter the Grand Prix with a partner. Finish in first place three times, then second place three times each. You should finish equal in first and receive 500 rings each.