Train tickets in France: Buy ahead or ok to buy when you get there?

"is there plenty of room on trains when you buy tickets like a day or two ahead?" - yes, some people decide to travel at short notice and buy tickets at the station and immediately get on the next train.
However buying tickets in advance can be a lot cheaper, you get discounts. Once you buy a ticket, you are commited to the train listed on the ticket, you cannot use that ticket for an earlier or later train.

You can look up times and prices, and buy tickets, on: https://www.thetrainline.com/
Compare the prices for tickets for 2 months in the future, with the price for tomorrow.

Thank you Chris!

Substantial price difference.

I buy directly from SCNF's sight. https://www.sncf.com/en/booking-itinerary/book-tickets

To check the price difference see what the price is for when you want to go and what it would be for something tomorrow?

You can save a lot of money but also lose flexibility. It's a trade off; you have to decide if it's worth it.

Ok, this is weird, I enter in July 28 from Paris to Bayeux, and it's greyed out. Does that mean that the train is not running?

More likely it means that tickets for that day aren't on sale yet.

Yes, since July 28 is a Sunday, it may mean the train is not running, or else just not yet for sale. This line has been under construction for the past few years and most of that happens on weekends. But they sometimes hold back the ticket sales and later release them if/when they're certain they'll be able to run the train. You might check again just a month ahead of travel.
Percentage-wise, the advance discount is significant, but euro-wise, these tickets are relatively affordable, even at full fare.

Weird, because they are for Paris to Rouen

Weird, because they are for Paris to Rouen

Well, that's a different line so you can't go by that. When I searched Paris to Bayeux on Sun July 28, I didn't see any morning trains running but there were trains running later in the afternoon. That leads me to believe they might have scheduled work on that line on Sunday morning.

Thank you!

I have the OUI.SNCF app on my phone and purchase most of my tickets on the app.