How do I know what season it is?

[Regrettucini]

Also, while I'm here, how do I check the date?

[lizardmang]

The date is displayed in the top-right corner (day of the week followed by day of the season).

The current season is displayed as an icon just below the date (you can find the icons here: https://stardewvalleywiki.com/Seasons ). The season is also very apparent by the music and appearance outside once you're familiar with them.

There is also a calendar just outside of Pierre's store that displays the current season's birthdays and holidays.