Will a satellite escape from the bounds of Earth's gravity if its orbital velocity is increased to escape velocity? If orbital velocity $(v=\omega r)$ is increased to escape velocity at that certain orbit, will it move to infinity? I know that it will skid from that orbit as vehicles do when their velocity is more than the equivalent velocity of banking of road. But this satellite has been revolving, shouldn't it just move to another orbit and revolve instead of moving to infinity since gravity is still effective on it?
 A: Increasing the speed will increase the orbit. At some point the speed corresponds to an infinitely large orbit, meaning that the object will never return. This is called escape velocity.
So, per definition, if an object such as a satellite - initially in orbit or stationary, that doesn't matter - reaches escape velocity, then it will not just reach a higher orbit but will never come back. By definition. If it did come back and just reached a higher orbit, then we wouldn't have called it escape velocity in the first place.
And to the note on gravity, be aware that gravity always acts on an object. Also on objects that move with escape velocity. The velocity is just large enough to outweigh the effect of gravity continuously.
A: If the satellite's new direction is such that it does not crash into Earth or feels friction from its atmosphere it will move to infinity.
The escape velocity 'at that orbit' meaning at that height is such that $$v^2/2 = GM_E/(R_E+h) \,.$$. This means that the binding energy, total energy excluding rest energies of satellite and Earth, vanishes if the satellite is boosted to escape speed. From then on it is an unbound system and dissociation will occur.
