Fantastic game. I know my way around it better than most, here's some tips to get you started.
Chaining involves sealing an enemy by shooting them with a command shot (or using command motion
+ B when within grabbing distance), grabbing the sealed enemy, then throwing it at other enemies. The seal will expire if it flies off-screen or touches terrain for too long, so be quick about grabbing it. Chaining is where most of your points will come from, so focus on it above all else.
Whenever you're not chaining, you should be shooting the seal for high hit counts (or 'pursuit', as it's referred to). Both for score, and because it will release a health power-up when you either start chaining again or the seal expires. Every 10 hits makes the power-up restore slightly more when it gets released. A particularly high hit count can bring you up from a sliver of health to a full bar!
Certain elements are better for chaining & pursuit than others. Ice is best for chaining overall, but Fire works slightly better for the first and seventh stages. Wind is the best for pursuit, but has the weakest normal shot, and chaining is much more difficult compared to the other two. Avoid using Light unless you're playing purely for survival; it's the strongest shot in the game, but is absolute shit for chaining or pursuing.
You can switch your current element one of three ways. Shooting up a formation of enemies with your normal shot (then shooting the yellow crystal so it changes color to the element you want), bombing, or using the 'discard' command shot (
+ C). Discarding comes in handy if you don't want to destroy any on-screen seals with your bomb attack.
Collecting teacups during Tea Time recovers your health. If you don't need/want the health, try dodging all the teacups instead. You'll get a secret bonus if you can pull it off.
Hope this helps!