How can I check the first date of a Steam game being played in my library?

Short answer, nope.

Play sessions are not explicitly tracked by Steam (or, if they are, this data is not accessible on a user level through either the API or an interface), as far as I know they only log the most recent time you've played it which is then overwritten each time you start the application again (while online).

Depending on why you would want this there might be other solutions.
What would you use it for?

The short answer is, Yes you can view when it was first played. This can be done from accessing the library, and the going to the game you want. For example I will take a look at beamng.drive. So you clearly see the last played date, To find out the first played you must go to the Activity bar, scroll down a bit till you find where you first played it. And so you can clearly see when I first put it on my wish-list then bought it and played it for the first time. Best regards, eShadow

Edit: Some games allow you to view this option from the in-game status menu. Others don't. If your game doesn't support it you can use my answer.

I know this is super late, but for certain games such as CS:GO, if you click on activity under your name, then look on the right side of the screen and click on games, some will have a button for "personal game data", and for CS:GO it will have your first time played displayed with exact time and date.