I've been learning the piano for 6-7 years.
Also, I played the saxophone in Instrumental lessons.
Technically, I know how to play the:
Voice (why not), clarinet, trumpet, recorder, flute, guitar, cello
I would definitely suggest a stronger reed. You can get some high notes out with a strength 2 reed but they probably won't sound too good and you will squeak a lot. I would suggest at least a 2 and a half but eventually you will have to go to a strength 3 reed to really get a good sound out of the high notes.
High notes require a lot of air from the stomach so just make sure you support those notes. Try to tighten up your lips when you play a high note so you don't squeak/go flat. Make sure you constantly keep your instrument warm as you practice so your notes don't sound dry or airy.
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Adding onto this, as with any wind instrument,
don't use your 'chest' air or use your throat to push out air. You'll only strain it and possibly damage it.