Tongue trills or lip trills?

For our purposes here, they are interchangeable! Do whatever feels the most relaxed in your instrument.

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Does it matter if I breathe in through the nose or the mouth?

It does! But not that much. Breathing in through the nose is optimal for most people, but please listen to your body. If you find that your nose is tight or doesn't draw in much air, please experiment with mouth only, or mouth + nose. Over time, you can move to trying just the nose. But really...it's splitting hairs! Listen to your body and do what gives you the best breath. Then, build off of that.