DC Relays do not take high inrush currents when first switched on. To a good approximation, the relay coil looks like a inductor in series with a resistor to the driving circuit.
The small movement of the switch does change the inductance slightly, and makes a bit of energy in the inductor appear to go away from the driving circuit point of view. However, these effects are both quite small relative to the inductance and resistance of the coil. In practice, you simply ignore this effect.
The inductor exhibits the opposite of inrush current, more like a soft start. When first turned on, the current starts at zero and rises linearly. The actual current profile over time is a exponential asymptotically approaching the applied voltage divided by the resistance.
The thing to watch out for is switching the relay off. The inductor current will not go to zero instantly. In the short term, the same inductor current will flow immediately after switch-off as immediately before switch-off. The inductor will make whatever voltage is required to keep the current flowing, which will be big enough to fry the switching transistor unless you give the current a path to flow.
A diode in reverse across the coil accomplishes this. When switched off, the inductor voltage only needs to be high enough to turn on the diode. After that, the inductor current will decay due to the resistance of the windings effectively in series with it, and the forward drop of the diode.
When particularly fast off-action is required of a relay, you add a resistor in series with the diode. You know the relay coil current when on. You size the resistor to get the largest voltage drop across it you can tolerate without damage.
AC versus DC driven
What I said above applies to driving a relay with DC. As Tony Stewart pointed out in a comment, things are a little more complicated when AC is applied to drive the coil.
The problem is that the material in the coil core can exhibit magnetic hysteresis. This means the property of the coil at turn-on depends on what state it was in when last turned off. If the core is left heavily magnetized in one direction, then the inductor that is the coil will hit its saturation current limit much sooner in that direction than normally. If you happen to switch on the coil at the right point in the AC cycle, the inductor can saturate. This means the inductance gets very low.
However, the resistance of the coil is still there. The worst case current is the peak voltage of the AC waveform divided by this resistance. The result is more current at startup, but not drastically more. Even AC relays still generally rely on the coil resistance to set the current. The inductance will reduce the average current somewhat, but usually not too much at low frequencies of the power line. This is why AC relays are specified to work up to some frequency. Above that frequency the inductance of the coil reduces the average current too much for the relay to reliably operate.