The tourist takes a camera and he's recording a video.

It is fine. Yes, you could drop the first "video" beacuse the second one will explain it. Besides, most photo cameras can shoot video these days so it's less of an issue. You could even drop "and": The tourist is standing with a camera making a video. Some variations: - The tourist is holding a camera and making a video. - The tourist is shooting a video with a camera. - The tourist is shooting a video with his camera.