Yes, you do have to fly the heading they give you, because they'll use that as the basis for any corrected vectors they assign after that. For example, if they say "Fly heading 270", you fly a heading of 270°. If they look on their scope and your track isn't taking you where they want, they might say "turn right 10 degrees", but in my experience, it's equally likely that they'll say "fly heading 280," which still means "turn right 10 degrees" (since they're assuming you're currently flying 270° by your compass).