John hopes...

“John has been learning a new musical instrument. Tomorrow he’ll give his first performance.”
How to put this while using the word “hopes” : “John hopes he’ll perform well tomorrow.” Correct ?

That’s good, yes.

The expressions are fine as three separate sentences, but here’s one way of combining all three statements into one sentence:
Tomorrow, John hopes he’ll go well when he gives his first performance on the musical instrument he has been learning.