They used "weather ropes", which was a rope hung outside every night before bed. When they woke up, they would bring the rope inside and look at it. Wet meant rainy, cold meant cold, hard as a rock meant really cold, and if the rope was missing, that meant it was windy.

Edit: ok, that was fun, here's a real answer: https://academic.oup.com/edited-volume/43505/chapter/364127476?login=false

tl;dr: they made observations and created almanacs. A combination of the two was usually "good enough". No mention of war in anything I was able to find, but I'm sure it was just as large a factor as it's always been. The Art of War was written like 5,000 years ago and talks about considering the weather, so it was definitely something humans had been doing for a while.