Why is there no electric field in the ideal wire? Why is there an electric field only inside the resistor in the circuit (assuming that the wire has no resistance)?
If there is no electric field inside the wire, why does the electron move to the positive pole?
 A: There is no electric field in a zero resistance wire, but with the exception of superconductors, all real wires have some resistance. In a superconductor electrons can freely move with no energy loss and therefore require no electric field.
The resistance of wires in a circuit is typically so low compared to resistors in the circuit that they are generally approximated as zero resistance conductors for the purpose of circuit analysis.
Hope this helps.
A: The wire has the same potential everywhere if its resistance is zero. There is no force needed to keep the electrons flowing in it because of the very fact that its resistance zero. The positive pole has the same voltage as at the resistor. So inside the wire, there is a constant voltage and no electric field. But how can the electrons flow? What gives them their motion? They are kicked into the wire and battery pole by the inside potential and beacause the wire has no resitance they continue to have this motion. They are not slowed down by the resistance of the wire, as this is zero.
A: 
Why there is an electric field only inside the resistor in the circuit (assuming that the wire has no resistance)

There is an electric field inside a resistor, or a capacitor for that matter is because there exists a potential difference across it. The potential difference exists because the battery deems so.
The wire doesn't have any potential difference across it -its an equipotential, an ideal conductor with zero resistance and hence can't hold an electric field inside.

If there is no electric field inside the wire, why does the electron move to the positive pole?

Oh no. The electron moves precisely because its experiencing an electric field - that created from the battery. Where does the electron go after leaving the negative terminal of a battery in a circuit? In the connected ideal wire? No. It goes straight to the resistor, then to the next component with the field.
But then why would one draw wires in circuit diagrams at all? That's because in real world circuits charge carriers do travel through wires to other components - they can' t jump through the intervening air gap$^1$ and need a conductive path. We carry forth this intuition in the form of an idealization while drawing circuit diagrams, drawing finite length ideal wires between circuit elements and showing currents even though, for this idealized diagram, the wires don't develop a field inside and play no role beyond that of showing connections.

$^1$ at lower than breakdown voltages
A: It doesn't take force to keep up motion. It takes force to generate motion. This is Newton's 2nd law.
Only if there are frictions, resistances and the like along the path will you have to exert a forwards force to maintain the motion. In an ideal wire, there are no sich resistances or frictions. So in an ideal wire, no electric force is needed. Charges will continue moving with constant speed effortlessly.
No wire is ideal, though. So no wire contains zero electric field. But it can be very small. The closest to zero is within superconductors.
