Red sky at night, shepherd's delight
This one holds up in the mid-latitudes, where weather generally travels west to east. A red evening sky means the setting sun is shining through dry, dusty air to the west — the air that is heading your way next.
A red morning sky means that dry air has already passed over you, and what follows may be wetter. Rhyme, but also reasoning.
Ring around the moon means rain
A halo around the moon is caused by ice crystals in high cirrostratus cloud. That cloud often runs ahead of a warm front, and a warm front often brings rain within a day or so.
Not reliable enough to cancel anything. Reliable enough that watching for it is a decent habit.
Pine cones close before rain
Genuinely true, and for a dull reason: the scales absorb moisture and swell shut in humid air. A pine cone is a small, accurate hygrometer that costs nothing.
Cows lie down before a storm
Charming, and not supported. Cows lie down for many reasons, chewing being the main one. If you have ever watched a field of cattle on a bright still afternoon, you have seen the counter-evidence.
Sound carries further before rain
Often true. Under a temperature inversion, sound refracts back toward the ground instead of dispersing upward, so distant church bells and motorways seem suddenly close. Inversions often precede a change.
Aching joints predict weather
Contested, but not absurd. Falling barometric pressure may affect joint tissue in some people. The studies disagree with each other more than they disagree with your grandmother.
Swallows flying low
Partly true, and secondhand. Insects fly lower in humid, heavy air, and swallows follow the insects. The birds are not predicting anything. They are just eating where the food is.