If you want more rings, then it is not hard to acquire rings fast, if you want, then you have three choices, arranged from the taking more time to the taking less time :
1. Play Grand Prix Mode and collect rings on the track, plus 500 rings for winning the cup in first place.
2. Buy the Gambler Extreme Gear, that is sold for 5160 rings, then use it on tracks (or The Grand Prix Mode to collect with it more rings) to collect more rings by installing the two ring caps. Installing them will let you collect 250 Rings instead of 100.
3. Complete the Crimson Crater track in Heros story mode, then choose mission 2 on the same track (Crimson Crater). Your goal is to collect 130 rings or more in one lap. It is easy and it is the fastest way to collect rings.
Your best choice is the third, because if you collect 200 ring per lap, for example, you may get 1000 Rings in only five laps (or by repeating the mission five times)