This is a clear example of weather lore that's been around for centuries and it's actually true. According to Wikipedia, "If the morning skies are red, it is likely that clear skies to the east permit the sun to light the undersides of moisture-bearing clouds coming in from the west. Conversely, in order to see red clouds in the evening, sunlight must have a clear path from the west in order to illuminate moisture-bearing clouds moving off to the east."
While many of the tales regarding weather prediction are not accurate, others do have a scientific basis.
"No weather's ill if the wind is still" - Winds have been used to predict weather for the longest time. Eastern winds typically announce the approach of a low pressure area (stormy weather), while western winds forecast clear and fair conditions that should remain constant for at least a couple of days.
"Circle around the moon, rain or snow soon" - A halo is a whitish ring that forms around the moon. It happens when the ice crystals in cirrus clouds reflect the light of the moon, and it's a sure predictor of either rain or snow.
"A summer fog for fair, a winter fog for rain" – Fog forms when the air is cool enough for vapor to condense rather than evaporate. In summer, the sky must be clear in order for the air to be cool. Winter fog forms when ocean air moves over land, carrying humidity with it.
"Rainbow in the morning, need for a warning" – Rainbows that form towards the west in the morning indicate rain is imminent. A rainbow that forms in the evening usually mean the opposite: Rain is exiting and fair weather is here.