Every time the ship goes into Hyper Mode is a moment of awesome really.
The Japanese seemed to like the ship enough to take all those pictures when it came down.
Radar operator: Japanese air force report sightings of the aircraft above Tokyo, sir.
Radar operator 2: Japanese air force reports the aircraft has left Japanese airspace.
Dr. Faraday: Where's it going now?
The look on Faraday's face right after saying this, as he watches the ship moving away from Japan at a ridiculously high speed.
In a meta sense, the transformation scenes of the ship were bleeding-edge CGI of the time. Even 30+ years later, they still hold up fairly well.
The scene where David enters the NASA hangar and first encounters the alien ship. From the moment he first sees it, to the first time the ship morphs its hatch into the levitating steps is totally awesome.
David's mom gets a minor one when David calls the house (under NASA surveillance) from a payphone. Thinking quickly on her feet, she loudly tells Jeff not to hog the phone in case David calls, buying him a few suspicion-free minutes to arrange things with David.