Amtrak can get rather expensive during the holidays and even more so last minute. Even with such a short trip. Commuter rail will at least be the same price regardless even if it’s a tad longer.
I’d recommend the Amtrak (faster) although it may not be easy to get seats together if going with kids. If you want to drive, use SpotHero app. It will be a lot cheaper than rolling into a garage (I’ve gotten $10 for a day).
As long as you check the times / schedule and can show up, you don't need to buy in advance. The only time I've seen a train to Boston on a Saturday be super full was the morning of the women's march in 2017.
JohnMellencamp21 t1_ivv0mdo wrote
Take the commuter rail and buy a ticket on board https://www.mbta.com/schedules/CR-Providence/timetable