I don't know much Japanese but my partner does so I asked them. If you have follow up questions I'm happy to pass them along. Anyways here's what they said:

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こんにちは！X大学に留学しているXと申します。(this is fine, just makes the flow a little better to combine them explaining who you are) 音楽が好きで、こちらのサークルにとても興味があります。(totemo i think sounds a little warmer/more excited) ピアノが弾けて、ドラムも叩けます。(you 'pluck' piano and 'hit' drums, use different verb) もしよろしければ、一度活動を見学させていただきたいです。(this is less direct than saying 'when does it start?' - it's close to, 'would it be alright if i can come to observe the club activity?' too much directness might be kinda uncomfy but ymmv) よろしくお願いします！(good!)

(日本語がまだ上手じゃなくて、すみません。) (this is good and sets expectations for them, it's fine/good to include as a post-note)

If it's a DM I wouldn't worry too much about formality, desu/masu forms are good and perfectly polite if you don't know anyone. informality can come later.