Why is it incorrect to say (Someone) 想要了一 (Something)? Basically whole question in the title. I was told that saying 我想要了一杯茶 or anything along those lines is incorrect. Why?
I think the problem is the usage of the character "了".

When "想要" and "了" are used together, the object is usually placed between them.  e.g. "我剛才不渴, 但現在想要一杯茶了."  The object may be omitted too, e.g. "親愛的, 我想要了."  (....the wife whispering to her husband....)

You may also have noted the difference between "想要" and "想要....了".  When "想要" is used alone, it refers to the intention/though/request itself.  When "想要....了" is used, it usually signifies a recent change or a decision just made.

As for "想" vs "要" vs "想要", cingS has given a clear answer already.