I would get the Hoppa. I know we give it a (justifiably) hard time here, but it will leave from a marked stop at the terminal and take you to the front door of the hotel. It leaves from right outside the T3 front door.
The local free buses (free only to the actual airport boundary, one stop beyond and it's the normal £1.50 fare) are fine if you know what you are doing, but it sounds like you are a visitor.