Aim carefully when flinging the animals at the end of a run. If you can place consecutive animals in the basket, each subsequent one will earn you 100 more points. If you miss, they reset back to being worth 100 points. If you can land them in the basket quickly, your character will wait longer before leaving the screen giving you more time to earn extra points.
Each time you increase your Fever level, you will move up the ladder of Fever patterns. Each pattern has a different distribution of rings and point bubbles. The higher the level becomes, the more those patterns shift towards point bubbles instead of rings. This will eventually noticeably decrease on how many rings you can collect per run. If you trigger Fever more than once in a run, you can see what the next pattern looks like. If needed delay upgrades until you have collected a large number of gold rings.
Rings, points, and collectibles are placed in such a way that trying to collect all of them in a run will not be efficient. Focus on collecting one type of objective per run. For rings, equip a ring-collecting Hero or Neutral Chao, then concentrate on grabbing the rings and triggering Fever mode. For points, equip a Dark Chao and focus on enemies. Work on extending your combos to trigger Fever mode as many times as possible. Also free your animal friends for a score boost. For collectibles, use the Chao that rescues you from falling and avoid long combos, because you can miss items if you enter Fever mode without collecting them first.