Go for a walk.
No, really, go for a walk. But notice the date on the calendar first, and if you can, notice what phase the moon is in.
And while you are walking, notice things. Notice which flowers are in bloom. Whether the trees have leaved yet. How far they have gone. Whether you can see the moon. Well enough that you can describe this in prose.
Then do it again next week. And the week after that.
While you are doing it, remember that in the vast majority of societies, everyone would notice all the details you notice for good reason. They lacked calendars, or at least good ones. Their society might have good calendars in places, but the slowness of communications might hinder their spread. At that, the questions that most concerned most people could not be answered by calendars.
A farmer instead knew the proverb that it was time to plow when flocks of grackles settled on the field and linger. (Because they could find grubs to eat, because grubs were rising in the earth, because the ground had thawed out.) Watching for signs of harvest, or bad weather. Considering whether it was safe to make hay given the danger of rain indicated by a ring around the moon.
He had an abundance of guidance in proverb form. The National Weather Bureau studied proverbs in the beginning of the 20th century, and concluded that all proverbs that ascribed particular importance to calendar days, such as Groundhog Day, were unreliable. Otherwise they were as good as the bureau at the time.
Gentry, nobility, and royalty needed to be aware of the weather as well, if not quite so omnipresently as the peasants. Perhaps. Sufficiently low-ranking people oversaw, or even worked at, agricultural tasks as well. At the very least, conferring with the grooms about when to let the horses out to pasture, and to breed them, and when hunting was feasible.
At that, other activities had to be finagled in around the weather and seasons. Winter snow and spring floods interfered with travel. Rain in any season could make the roads impassible with mud and flood. Fairs or courts held during sowing or harvest would mean famine.
The higher ranking you were, the more important the transportation was. The peasants might never use anything not found or made within a day's journey of their homes. Nobles enjoyed far-fetched luxuries. Indeed, their status might turn on how well they could supply guests with such luxuries. They might even be required by sumptuary laws to wear imported cloth.
Worst of all, you might have to go to war in bad weather. Wrangle horse, foot and artillery over the sodden roads and through the flooded fords. Trudge through heaps of snow. Fight on a field that's more marsh.
The one offsetting virtue was that your foes faced the same conditions, more or less. The advantage was to the defender because he did not have to move to so far, or transport food over such a distance. He would have to worry about the well-being of the peasants while gathering food for a siege, but the attacker had to worry about bringing food in if the defender had prudently removed it.
By the same token, travel into new climes is noticed. Your characters will quickly notice the increasing cold from climbing mountains, or the tempering effects of the seashore (or large lake) on both hot and cold. Going south or north will have them noting the changing start of summer, or winter, and even consider the effect on agriculture.
If someone from England ended up in Greece, he might write in astonishment that their growing season was the winter, when it rained, and not the summer, which was drought-stricken. (The original account of the myth of Persephone did not mention what season was dead because of her descent to Hades. More northerly climes of course assumed it was winter, but it was probably summer. Leaves were out, but who cared about leaves? You can't eat them!)
You could keep track of these kind of changes by the sun, the moon, and the stars mentioned up front. You kept track of those. Indeed, in planting and harvest, you would work by the light of the moon in order to get the seeds planted or the harvest in, in the time you had.
If your world has characters who would notice all these things, their viewpoint should hold them all. Thus, you need to know them.
Yep, yep and a lot of yeps.
Many of my neighbors in the early sixties, Athabaskans, Eskimos, told time that way; "See those clouds? Goose season!" , "No snow left of the side of Mount Biggun, time to move to fish camp, salmon will be running soon."
Also I found the weather saws from the lower forty eight, if I just moved them a month, still worked. April, not March, comes in like a lion and goes out like a lamb. May showers bring June flowers.
I suspect if when we inhabit any other rotating planets we'll still be working the old world weather saws into the new; "Red sky at night, the rail runner's delight. Red sky in the morning the runners take warning."
My parents' generation (Dad born 1928, Mom 1931) knew how important all of this was. Another sign of what time of year it was, according to them, was what sort of illnesses began to pass around. Some sicknesses only showed up when summer was 'here' and others were restricted to winter.