a ball throw vertically up moves under the effect of gravity and eventually falls back down after reaching a certain max. height. what is the direction of velocity of the ball on the way up and down and what is the direction of acceleration of the ball on the way up and down

While going up, direction of velocity is upwards and velocity is decreasing with time due to gravity, which is pulling it down.

While coming dowm, direction of velocity is downwards and velocity is increasing with time due to gravity, which is pulling it downwards.

For motion under gravity, acceleration vector due to gravitational force will always be downward, whatever be the direction of velocity.