"I want coffee. I do not want milk."
This is correct, but not correct enough. 我想咖啡 would be like "I think coffee".
想, when used alone, is almost always intransitive. With only two exceptions:
The object is a person (or more), where 想 means miss: 我想你 = I miss you.
The object is the word thing, or other similar abstract idea, where it means think and has the same usage as "think" in English: 我在想为什么他没有来 = I'm thinking why he didn't come.
When 想 is followed by a verb, it means want to + verb.
我要咖啡 = I ask for coffee. 我想要咖啡 = I want to ask for coffee. A more polite expression.