Our feet start moving first when a bus starts suddenly because they are in contact with the floor of the bus, which moves with the bus.
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When the bus begins to move suddenly, we tend to fall backwards because the bus and our feet in contact with the floor start moving, but the rest of our body opposes this motion due to inertia.
Safety belts exert a force on our bodies to make the forward motion slower, thus preventing collision with the panels and reducing the risk of injury.
When the driver applies a braking force, the car slows down, but our body tends to continue in the same state of motion due to its inertia.
We tend to remain at rest with respect to the seat because of inertia, which is the tendency of an object to resist changes in its state of motion.