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.
@ccless1 have explained quite good, but there are some problems unsolved.
想要了 is used in Chinese actually.
想要means I would like something
要means I want something
I would like a cup of tea, please
give me a cup of tea, b***h
By the way
我想一杯茶 is not actually proper Chinese for I would like a cup of tea
想 have the meanings of think, or miss
我想一杯茶 means I am thinking about / of a cup of tea
In this sentence like you could use either 想要，想 or 要，as all mean "to want," though you should note that there is a slight difference here in terms of tone and politeness.
我想一杯茶 means "I would like a cup of tea."
我要一杯茶 and 我想要一杯茶 mean "I want a cup of tea."
The link below might be helpful.