Certainly you can use several linking words to create a long sentence. This is the primary way sentences become long, in fact. However, please note that this often makes the sentence less clear/readable than having several shorter sentences that are each simple, direct, and clear.
As far as your second question on comma usage goes, I'd have to say that to a large extent it depends on the details of the sentence. In your example sentence, I'd put a comment after the first clause (i.e., before the second linker), because of the details of the sentence content:
"I want to improve my piano skills and play in groups with other pianists, as well as meeting other young keyboard players."
This is because the first linker links two objects of the verb "want" that form a list, while the second linker connects a secondary clause. If you'd phrased it as a three-item list, you'd use commas as a normal list, e.g.:
"I want to improve my piano skills, play in groups with other pianists, and meet other young keyboard players."
In that case, I personally prefer the Oxford comma before the last list item, but that's a personal preference not a requirement.
In short, comma use comes from the details of the sentence structure, not from the simple question of whether or not there's a linking word.