Learning the levels, which characters are best for levels, super star locations, and when to pause is huge.
Levels are self explanatory. If you know where it is safe to jump... you'll be fine. Timing can be an issue for stringing some skills/combinations together. Knowing the level well enough to string together long chains of jumps off multiple enemies will build your star power quickly. Though, I've noticed double jumping with no object/enemy/coin nearby still yields a little "star power contribution" indicator under Mario/Toad/Luigi/Mrs Toad (those are the ones that can double jump, right?).
I understand I am very bad about landing on spikey things. Therefore, when I see sand worlds, I tend to favor Yoshi since he can land on them without penalty. I also rely heavily on double-jumps for stringing together combinations and Yoshi/Princess make that difficult for me. Personally, I feel like shrinking and losing coins (Mario/Luigi) is more beneficial than immediately getting bubbled and taken back in the course.
Any time you see a red ring that spawns 5 red coins, GET IT. Well... get the coins too. This will result in a super star/coin rush proc. You become invincible and a coin-magnet. Someone help me out if I am wrong - but - the more full your star power gauge is the longer your star-power lasts. I've also notices that things that add star power to the gauge continue to do so. Just not at a rate faster than losing star-power.
In my opinion, the hardest thing to learn that ups your win rate is: Learn when you should pause! Nothing is more mentally derailing than being 1 jump away from getting your coin rush (or just procing your coin rush) than immediately falling off a platform/jumping into a fireball/getting hit by a flying wrench instead of taking half a second to let the danger pass and racking up a score.
I think another thing that influences your toad-gaining-count is the "total toad difference" between you and the person you are challenging. I've seen bigger toad gains when I had 400 and the person I was challenging had 1000+. I do not know if there is a secret differential for max gains.