Alright this is what I've gathered from playing and reading other people's tips on here, apologies if I make some mistakes as it's a work in progress.
Points:
Calculated at the end of each segment.
Perfect turn = 200 each
Yellow strip = 100 each (up to 300 each with 3x multiplier)
Blue ring = 100 each (up to 300 each with 3x multiplier)
Health note = 300 each if you're undamaged (up to 900 each with 3x multiplier)
No Misses = 1000 (+500 for each additional segment completed with no note misses)
No Damage = 1000 (+500 for each additional segment completed with no note damage)
Kills = 50 each (sending a projectile wave by ground pounding or attacking the boss to kill "tentacles" that appear in the background, see below in Other tips)
You don't actually get any points for simply hitting notes and turns. Hitting notes does raise your base multiplier though that appears in the top right of the screen. 8 notes hit raises the multiplier from 1x to 2x, and another 8 raises it to a max of 3x. This multiplier affects health notes, yellow strips, and blue rings as seen above. Hitting turns do not affect the multiplier and perfect turns aren't modified by it.
Oh yeah, no points for hitting the yellow strips without having A/X held after a note hit either.
More in-depth:
If you hit a health note while you're undamaged, you get extra points as indicated by the little blue dots that appear on the side of your beetle. You'll also notice these dots also come up whenever you hit the yellow stripes as well as the blue rings, and the number of them that appears also goes up with your note multiplier in the top right. Each strip and ring is worth 1 blue dot, and each health note worth 3, and the amount gets tripled with a max multiplier of 3x if you're not missing notes. Each of these dots - which are diamonds - are worth 100 points and you see them at the end of the segment in the point total. They are your main source of points.
Starting at a multiplier of 2x, you can create health notes by hitting Up + holding A/X on one note, which will start and maintain flight, and pressing Down to land on another. This will turn the next upcoming note into a health note. You can create one health pack with just two notes, but at times you may want to extend flight over multiple notes or turns before coming down. Extending flight can be done by hitting perfect turns or quickly tapping Up + A/X on notes to hit them as you pass.
In conclusion:
For the most points you want to try and hit things so you're generating as many health notes as possible during the segment while not missing rings or strips, since if you miss 4 or more of those in trying to make a health note you're looking at a net loss of points. It goes without saying that you don't want to miss notes or take damage at all either, and reverting to a checkpoint will reset all multipliers. If you are going for a highscore in a level and take damage and miss a note, then at that point it would be more beneficial to reset the checkpoint and play as perfectly as possible.
Other tips/bits of knowledge:
-There are "tentacles" that creep up from the background in certain parts of each level that can be killed by performing a ground pound on a note or attacking a boss to create a shockwave (you don't need a multiplier going, and their appearance depends on the level asthetic). The pulse from these actions will travel along the track killing them and making a scream sound signifying kills. Each one killed = +50 points to your score total. - credit to Genkobar
-You can also create health notes during bosses. After a successful sequence of not getting missing notes/getting hit, attacking the boss will spawn a health note. You can spawn a second health pack if you pound down after flight on the attack note. -credit to jett, epmode, and EVO
-On later levels you'll also see yellow strips come directly after blue rings with no note in between. Just press Down immediately before coming to the stripes and you'll hit them for points.
-The S-rank in the leaderboards isn't necessarily associated with the score next to it. It just means the player has S-ranked that level at some point, and their highest score regardless of rank is shown next to it.
Other things I still don't know:
-If you pound a health note, when it goes into slow motion, I don't know if that's generating a significant amount of extra points, or it just looks cool/messes up your rhythm. Probably both?
-Boss battles. Some things might work differently there since you can extend them indefinitely if you miss notes. I'm guessing they designed them so you can't up your score this way infinitely, like putting a cap on perfect turns, but I don't know.
-And some other stuff I'm sure I'm forgetting.
