A circuit and a magnet in a relative motion First:
You have a circuit and a magnet. Will there be any difference in the induced current that flows in the circuit If


*

*(a) Circuit is at rest, magnet is moved towards the circuit.

*(b) Circuit is moved towards the magnet, the magnet is at rest.


There is no difference because of the principle of Galileo's relativity. (or because of another reason, but a and b are very easy to understand, intuitively.)
Question:
What if we move the circuit and magnet in the same direction and the same speed, will there any induced emf in the circuit? Moreover, what if we we just move them at the same speed but not the same direction, will there be any difference (so does direction effect the result here?)
Is there any relationship between this question and the special relativity principle (like in the moving wire and moving particle.)
 A: 
What if we move the circuit and magnet in the same direction and the same speed, will there any induced emf in the circuit?

There is one case where the electric conductor and the magnet move together and an EMF is induced. This device is a homopolar generator, also called a Faraday disc

The amazing thing is that not only a driven ring induces a voltage, but


If the magnetic field is provided by a permanent magnet, the generator works regardless of whether the magnet is fixed to the stator or rotates with the disc. From Wikipedia


To be precise, in the above described case the conductor moves in a circle and by this the influenced by the magnetic field electrons are under the influence of an acceleration too. Moving the disc together with the field on a straight line, no emf will be observable. Moving such a device on a zigzag trajectory one induce an AC current.

Moreover, what if we we just move them at the same speed but not the same direction, will there be any difference (so does direction effect the result here?)

For cases without curved trajectories the only phenomenon you have to take in account are the interdependencies between electric current, magnetic field and movement of the electric conductor. One of this dependencies is the Lorentz force, where a current carrying wire in a non parallel to the current applied magnetic field will move the wire perpendicular to the plain, formed by the directions of the current and the magnetic fields:

For your case of the induction of a current the conditions are the same. As long as you move a wire in relation to the magnetic field (or vary the magnetic field of an electric coil) AND this relative change in parameters happens not in directions, parallel to each other, a current will be induced.
To see the direction of the induced parameter one has to use right hand rules, which illustrate how the induction process happens in nature. As said above, the induced parameter has an direction of action and this direction is always perpendicular to the plain, formed by the two incoming parameters.
Of course occurs the question, in which direction of the plain the induced parameter acts. Because as you could image, were are two possible direction (because a plain has two surfaces, well?). The direction of action was found out empirical, means during experiments and than proved many times. After it became a low. And it turns out, that involved electrons and anti-protons show a opposite behavior in relation to positrons and protons:

All images from Wikipedia.
