I learned history on family vacations - where we stopped at every historical marker and plenty of museums, other travel opportunities, and picking and choosing books that piqued my interest at a given time... The single biggest flaw/challenge with teaching history is putting it in context - the 'why' of history, i suppose. I'ts easier to place context when you are standing on the roadside, reading a historical marker overlooking a valley tracked by immigrant trails of the US Western Expansion, or perhaps trying to understand the setting of a semi-historical movie, or attempting to better understand the relationships between cultures and religions that extend into modern times....
Onto context - here we are gaming on our typically faux-medieval milleu. Not having been in an ongoing campaign - how much does weather and climate really enter into most games? There are certainly plenty of tables for randomizing weather, but how often do DM's crack them, and the associated benefits and challenges of travel and adventure during different seasons?
Enter The Little Ice Age: How Climate Made History 1300-1850 purchased used at the local Goodwill for the princely sum of $2.99.
Briefly, the book outlines and discusses the period of climatic variation/instability between the earlier Medieval Warm Period and the beginning of the Industrial Age. A number of factors lead to generally colder and wetter conditions in the Atlantic Ocean and Europe, with associated stress on culture and populations (there is limited information from other portions of the world, so the book focuses on Europe) .
Different regions and leaderships took differing strategies (if at all) in the face of the unknown -
The English and Dutch became more flexible and innovative in their land use and food production, while the French maintained more traditional feudal-era practices and management. This led to certain resiliency on the formers' part (although not completely effective) and contributed to increased famine and the eventual French Revolution for the latter.
The economically and nutritionally important cod fisheries of the North Sea collapsed due to decreasing sea temperatures and pack ice - the fishermen followed the cod east, into more distant and dangerous fisheries (there is somewhat credible evidence that Breton and Irish fishermen fished cod off the North American coasts in the 1400's and were perfectly aware of a continent there, but like any good fishermen, were bane to give up the location of a good fishing hole....)
Glaciers advanced from the Alps, blocking passes and destroying or putting towns at risk.
Landholders experimented with new crop rotation and breed adaptation. New crops were adopted, including the potato from the New World. Followed by the risks of monocropping in the case of Ireland...
Of course, during this time, Europe also began its surge of exploration and colonization, the Renaissance and Enlightenment took place, and Continent-wide political upheavals and wars mapped out modern political boundaries. So there's that.... Instability can breed innovation and action...
As discussed above, the weather and longer term climate is a local/regional stress-driver. Do the characters experience restricted travel and movement, or effects on their health?
Can a bad winter (or series of bad seasons) spur an invasion or conquest for resources - land, food, etc. Or perhaps take advantage of a weakened populace and military?
Are the characters moving among other displaced populations, and becoming caught up in political instability or intrigue?
What about prices and availability of food, other consumables, and equipment. Does a small, impoverished are even want a few extra mouths wandering through, or are they firmly asked to keep moving?
Just some thoughts... You'll come up with more creative and appropriate elements to your world than I can.