A Red Giant, such as Betelgeuse, would have a spectrum similar to that of a tungsten-filament car headlight. If our Sun were replaced by a Red Giant star it would look blindingly bright and not red at all. It is a fallacy to thimk of a Red Giant looking like a setting sun, it wouldn't. The sky would still look blue because there would still be a strong blue component in the spectrum. Of course a Red Giant at the same distance as our Sun would be the size of a football at the end of your nose, and there would be little likelyhood of Earth's atmosphere remaining.