Being a pretty wide open topic, I can take this opportunity to go over how I deal with weather in my campaign world, and being that this year has been particularly brutal all over the US with snow and freezing temps, it might be a workable solution for just about anywhere.
Take a step outside just before your game starts. That’s what outside in my game world looks like
First off, this really only applies to areas of your campaign world that are not thematically set to a specific climate. Travelling through a jungle is likely going to be warm, traversing an icy glacier would be quite cold. However for the vast majority of gaming, it works very well. I usually allow a weather system (i.e., whatever is outside at the moment) to persist in-game for about a week. Past that I’ll follow a likely forecast. The thing about weather and why it’s so easy to deal with, is because it’s all chaos theory anyway, but on a large enough scale that we can predict movements on relatively short durations somewhat accurately.
The icy embrace of winter has been cruel indeed this year, regularly putting my adventurers in 2 feet of snow and fully covered roads far too buried to drive a cart along. Extreme cold days they’ve been advised to find shelter throughout the day as even a well dressed traveler can succumb to sub-zero temperatures rather quickly. They even ran into a halfling messenger that froze to death locked outside of their doors. Of course it was carrying a message begging for help from a town across an icy lake, but with all the snow and wind that day, it was too dangerous to attempt travel and they had to put it off.
Tough decisions brought on by the perils of winter snows. It’s funny, because the players rarely give a shit about the weather outside, and usually for good reason. Weather can be boring. Really 95% of the time it probably should be boring. Either overcast, light clouds, clear, normal everyday weather. In the rare event that you have a storm rolling through, it’s palpable sitting at that gaming table, and it makes the environment that much more believable. This brings an extra dose of intense realism when you start a session with heavy thunder and lightning in a downpour if a player literally had to run through that in order to start the game. Everyone is painfully aware of the numbing cold we deal with in the north, and having it occur as naturally in game as you find in the real world gives it the immersion factor a player might need to take it seriously.