If there is friction and everything stops on earth why does earth not stop due to friction? Why do planets revolve around the Sun if there is a force called frictional force? Why do planets not stop rotating while the objects in motion stop after a while? Why are they continuously moving? Why are planets not affected by frictional force?
 A: Note, I'm going to answer at an appropriate level. For a physicist this won't be exactly precise, and that's okay.
Why do planets revolve around the sun if there is a force called frictional force? Why are planets not affected by frictional force?
Friction is caused when two things rub together. If you had a microscope, you could see the particles in them "stick" together as they rub, and then get pulled apart, over and over again, quicker than you can follow.  This process means that each of them feel the other "pulling" at them.
(If that's hard to think about, imagine if you tried to run fast on a surface covered with chewing gum. Your trainers would constantly stick a bit every time you place or move them on the track, and you'd feel yourself being held back or slowed down by it. Friction is a little bit like that at a tiny scale - the particles stick and come free, but it slows them down a little each time.)
For example, when a driver presses the brake pedal in a car, the brake pad presses on the brake disk.  The surface of the pad "sticks" to the disc, then "breaks" away, over and over, faster than you can count, and this slows the disc  - and the wheel which the disc is attached to. The wheel now finds itself "trying" to turn slower than it would when it rolls on the road surface, so the rubber "sticks" to the road over and over, and as the wheels are fixed to the car, the wheels moving slower on the road means the whole car slows down. So braking a car uses friction at least in two places. (The pad and the road can't move, so only the disc and wheel can be slowed)
If there isn't much or any friction, then the sticking and unsticking doesn't happen, and then there's no friction and no "slowing" effect (or a much smaller effect). If there is ice on the road, or you put oil on your car's pads and discs, they wouldn't brake when you wanted,and that's why.
Not all slowing is due to friction. Some is due to other effects. For example, if you have tried to walk against a strong wind, the wind can slow you down or speed you up, and that's due to wind, or wind resistance. Also if you put your hand in a fast-flowing river, or next to a fast flowing stream of air,  you can feel the movement of the water or air trying to pull your hand with it.
Planets move in space. But there aren't any (or enough) of these things in space to affect how they move. There is nothing that they are rubbing against or sticking to, no air to resist their movement, no fast flowing clouds or air  nearby to pull them faster or slower. So the only movement they have is their movement round the sun. There's nothing to change it.
Why do planets not stop rotating while the objects in motion stop after a while? Why are they continuously moving around?
This is a surprisingly good question - not a "beginner" question by any means. To answer it properly needs an understanding of a Einstein's Theory of General Relativity, which explains how gravity and movement through space seem to work in our universe. So any answer needs to be a simplified version.
Here are a few different ways to explain it. They all give a different idea of how it works - pick one that you like.
Answer #1: Satellite falling towards earth
Moving in a straight line is what things do when no force or energy is applied. You could think of it as the "easiest movement for a lazy thing to do". So a "lazy" ball tries to move in a straight line unless something pulls at it.
A ball thrown in the air is trying to move in a straight line, but earth is pulling at it. So the ball falls towards the ground instead, until it hits something that gets in its way and stops it. But a satellite is going round earth so fast that as it falls towards earth, the surface of the earth "curves" away , so it falls a bit - and finds it is still the same speed and same height above the ground.  Repeat constantly - and you have a satellite orbiting earth. It keeps going, keeps falling, and the earth keeps being the same distance away. Earth orbits the sun the same way.
Answer #2: Gravity "bends" space
Ignoring the sun's own movement, the earth moves round the sun because the sun's huge mass "bends" space in a way that is hard to explain in plain English. Because of this, the earth tries to travel in the "easiest" direction, but near the sun, that's a circle (or ellipse), not a straight line, because space is being bent. So its actually travelling in the "easiest" way, but near the sun that's a circle not a straight line.
Answer #3: The earth is very slowly losing energy and getting a tiny bit closer to the sun over billions of years, but it's incredibly slow, so we don't notice it. This is the energy lost as the earth orbits
Can I leave this one as a heading? I'm not sure how to explain it simply! Maybe it gets the idea across anyway.
Answer #4: Unless something else slows an object down, it will keep moving. In the world around us, on earth, there are many things slowing other things down - but it's often invisible, so we don't really see it. So we just assume "everything slows down and comes to a halt". But in space that's not the case.
On earth, the air resists movement, friction resists movement, you can't fall through thick solid objects like concrete or the ground if it's in your way. So its almost second nature - everything around us slows down. So it's natural to ask why objects in space don't.
But the difference is that on earth most moving objects are in contact with something that can slow them. In space that's not so. Objects in space just aren't usually in contact with anything that's slowing them down.
A: For friction to occur there must be a medium which can exert the friction forces. For example if a raindrop falls through air there is friction against the air and it reaches some terminal velocity after a few meters. If a car rolls on a street there is friction in the bearings and rolling friction between the road and the tyres which slows it down.
In space, however, there is (almost) no such medium. The gas within the solar system, the so called interplanetary medium is so incredibly dilute, that the friction effects are usually negligible even on the scale of millions of years.
There is, however, some research that suggests that the dynamic drag (that is, friction) of the interplanetary medium may be relevant in some processes in the evolution for planetary systems, see for example the paper Dynamical Friction and Resonance Trapping in Planetary Systems by Nader Haghighipour, where an orbital resonance is reached when the friction with the interplanetary medium is considered.

For completeness, I should add there are other effects in orbital mechanics where friction is relevant. The most common one is related to tidal forces and, for example, the reason that we always see the same side of the moon (this condition is called tidal locking). With these effects the friction is within the orbiting bodies resisting the tidal deformation and slows down their revolution until it is in sync with the orbital rotation. This exact effect also slows down earth's revolution which causes the days to become longer on the scale of hundred of millions of years. By now our atomic clocks are so precise, that we need to adjust our time-keeping to stay in sync with earth's slowing rotation, that is we can measure the effect of tidal deceleration (and other effects) on earth's revolution on the time-scale of years (by comparing the time our precise clocks give to the astronomic measurement of the position of earth). There is even evidence for this in old sediment rocks that formed in shallow seas with tides (see this Wikipedia article).
Also, there is Poynting-Robertson drag, where dust particles are slowed down due to the net radiation pressure tangential to their orbits.
A: The answers here seem to address all but one of the questions -- Why do planets not stop rotating while the objects in motion stop after a while?
Note that there is no net external torque on the planet, at least not due to friction inside it, and hence its angular momentum is conserved. So long as the shape of the planet (moment of inertia) does not change, nor will the rotation (angular velocity). 
When an object moves, let's say a ball rolling on the floor, the friction due to the floor is an external force for the ball -- it is because of the floor, something "outside" of the system (the ball) and hence can produce net external force and torque on the ball, affecting its motion. 
However, friction cannot stop a planet from rotating as long as the shape of the planet remains same. This is because all the frictional forces on the planet (except for the negligible drag from bumping into a few stray H atoms in space) are internal forces. Hence, they cannot produce a net external torque on the planet and thus cannot change its rotation unless its shape (and thus the moment of inertia) changes.
A: Your premises are true on a human time scale but not on a cosmological time scale. It is indeed important to understand that every macroscopic process involves friction, or more technically, imcreases entropy. (Think of entropy as "unusable energy", as in dispersed, undirected frictional heat instead of concentrated, directed kinetic energy.) Allowed enough time, the universe will die a heat death, which assumes that the earth has stopped circling the sun, the sun has stopped circling the galaxy, and the galaxy has stopped circling whatever it circles, all due to some kind of friction.
This is a fundamental trait of nature: Moving forward in time increases entropy. There are people who define the direction of the flow of time by "it flows in the direction of increasing entropy", or perhaps even "the increasing entropy creates 'time'". The reason is clear if you run a movie backwards. You cannot re-connect the shards of a broken glass, you cannot put the exhaust fumes back in the motor to drive the car backwards, etc., even though nothing of that would violate any law of nature, except the one that overall entropy always increases, never decreases.
This circumstance is more fundamental than the typical middle school physics education acknowledges, where friction is treated as an annoying distraction from the underlying principles. It is not; entropy (as in friction) is one of those principles.
Now let's quickly consider your questions one by one.

Why do planets revolve around the Sun if there is a force called frictional force? 

Because the frictional forces are very small compared to the huge mass of the planets. The interplanetary vacuum is very good by earth standards, so there is not much to rub against. It's basically the solar wind (a stream of particles emanating from the sun), electromagnetic radiation, and a few rocks and pebbles that are tiny compared to earth.
The orbiting earth is also creating gravitational waves which could in a broad sense be considered a form of friction, but the effect is immeasurably small in our time scale.

Why do planets not stop rotating while the objects in motion stop after a while? 

The planets do slow down; one of them (Mercury) and many moons like ours are already in tidal lock with the centers of their orbits. They all will stop rotating eventually if and when the universe becomes old enough.

Why are they continuously moving? Why are planets not affected by frictional force?

They are affected. Their orbits do change through friction; the effect is just very small. The lifetime of a human or indeed of humanity is very small compared to cosmological times, so we don't observe it as far as the orbits of normal planets go. We do observe frictional slow-down on the rotation of moons and planets, as Sebastian explained, when we measure very carefully.
