How do velocity and acceleration change with time and why do they change with time?

It all comes down to Newton's Laws. Acceleration changes with time
only if the force or the mass of the object changes with time
according to:

a = F/m,

that is, the acceleration is proportional to the force divided by the
mass. The rate of change of velocity with time is the definition of
acceleration, so if the acceleration is not zero, the velocity will
change with time directly as the time integral of the acceleration.

Your question is rather abstract. Perhaps you have a specific example
in mind that someone could comment on.

What is velocity? Velocity is speed, with a direction specified. What
is speed? Speed is how fast an object is changing its position. An
example of speed is "10 kilometers per hour (10 km/hr)". An example of
velocity is "10 km/hr eastward".

   Then what is acceleration? Acceleration is to velocity as velocity
is to position; it is how fast velocity is changing. Suppose you're
driving your car at a constant rate of 10 m/s. Your foot is holding
the gas pedal down a certain amount. The speedometer just sits there
pointing at 10 m/s. If you then push further down on the gas pedal,
the car speeds up- it accelerates. With each passing second it adds
some amount of speed to its motion. If, for example, it adds 5 km/hr
each second, we say it has an acceleration of 5 m/s/s, or, in other
words, 5 m/s^2. The rate at which the speedometer is changing its
reading is a measure of the car's acceleration. Acceleration is a
change of velocity.

   Consider again your foot on the gas pedal. Holding it in one spot
maintains a constant velocity. Pushing it down a little further feeds
gasoline in faster and the engine works harder, accelerating the car.
Suppose that (ideally, disregarding realistic frictional losses) you
proceed to not just push the pedal down by some fixed amount, but
steadily push it ever further downwards? Then the rate at which
gasoline is fed into the motor not only increases, but the rate at
which it increases also increases. The acceleration itself will also
increase steadily. It's sort of like having a knob that controls the
strength of gravity. If you drop a rock from arm's length at sea
level, it'll fall by accelerating downwards at roughly) 10 m/s^2.
Turning up the knob would change that to, for example, 10.5 m/s^2, or
whatever. Turning the knob up steadily would cause the rock to
accelerate its acceleration. If you turn it up at a constant rate, the
acceleration of the acceleration will be constant. If the rock
initially accelerates at 10 m/s^2, and you turn up the gravity knob
steadily, increasing the acceleration by, say, an extra 10 m/s^2 with
each passing second, then its velocity will be increasing by 10
m/s^2/s^2, or 10 m/s^4.

   As for why velocity & acceleration change with time, look again at
my example of the car. An object will accelerate when *work* is done
upon it; that is to say, when kinetic energy is added to it by some
interaction. Burning gasoline releases chemical energy from the
gasoline and transfers it to the car's machinery. Work is done upon
the car by combusting the fuel.

   Of course, if you know something about Newton's laws of motion,
you'll know about the law of inertia: a body in non-accelerated motion
will continue in that motion 'til it interacts with another body. So
why does a car need to constantly consume gas to keep moving? Once
work has been done upon it, bringing it to some velocity, why doesn't
it just keep going at that velocity? It's because the car is
constantly being accelerated -in the opposite direction- by
interactions with air & the road. The car does work upon them, and so
loses kinetic energy to them. The car must perpetually accelerate
forwards just to maintain a constant total speed.

QED: One minor quibble. Work is not required to accelerate an object.
If a force is applied perpendicular to the velocity, no work is done.
An example is a satellite in a circular orbit. The object is always
accelerating toward the center of the earth, but it never gets any
closer and never gains or loses energy. You knew that, of course.

acceleration is the change in velocity divided by time, and velocity
is the change in position divided by time. acceleration is equal to
force applied times the mass of the object. so the only way you can
change acceleration of any object is by changing the force applied to
the object as time progresses or to remove mass from  the object. one
example is if you wanted to changed the acceleration of gravity (9.81
meters/second^2) you would need to changed the mass of the earth or
apply a force to the falling object, like a rocket. now because
acceleration is equal to the change in velocity divided by time all
you need to do to change the velocity is have acceleration, or
deceleration. even if the acceleration is constant the velocity of the
object is always changing because there is acceleration. some
equations that are useful for when the acceleration is constant are:
Velocity(final)=Velocity(initial)+acceleration*time  

Position(final)=position(initial)+velocity(initial)*time+(1/2)acceleration*time^2
(position refers to points on an imaginary axis either vertical or horizontal)

Velocity(final)^2=velocity(initial)^2+acceleration*2*change in the position

these equations are useful when the acceleration is constant. again
velocity changes if there is an acceleration of any kind, positive or
negative, and acceleration changes if a force is applied, although the
direction of the force is important.