The only time I consistently think about physics is when I'm riding the train. Either I'm standing with nothing to hold onto, or I have one metal handhold amidst a jumble of arms and hands and elbows. As the train beeps and the doors slide close,
"stand clear of the closing doors please"
I have to think quickly: which way is it moving? Acceleration in one direction means I will be moving in the other due to inertia; to combat the movement and stay in place, as most sane people in public will try to do, I need to apply an equal but opposite force against the train's acceleration direction. This involves a combination of planting my feet away from my body against the direction of the train, exerting force through my shoes in that same direction, and if applicable, pushing or pulling (whichever way necessary) against the metal handhold in said direction. When executed successfully, you can reach a perfect state of opposing force, rendering you stationary. Congratulations! - you are the master of train physics.
As the train reaches a steady velocity, no amount of force is exerted on the car, and subsequently on you, so you may stand perfectly still if you trust the subway infrastructure (no recommendations here). The safe play is to be ready for any inevitable bump, speed, or twist that will arise. I sometimes like to stand freely with no handhold, trusting my core and reflexes to keep me balanced. As the train approaches the next station, the opposite happens from the original departure - despite the train moving in the same direction, force is being applied against the direction of travel. Due to inertia, you will be moving faster than the train in the same direction, so you will need to apply force in the same direction of travel this time to stay in place. Perfectly stationary.
The final part that catches most people off guard is the "settling" of the train after it comes to a halt. The train slows from moving to nonmoving, and right before coming to a complete stop it jerks (on a spectrum to slides) back in place, which is another flip in force. If you were still applying force in the direction of travel, then the final jerk will throw you further in that direction.
When train physics are improperly observed, and you get one of the train's forces backwards, then no amount of apologizing will excuse you for flying headfirst into the buxom lady next to you!