Using a car powered air compressor with Arduino

I am currently working on a project for an inflation/deflation system and am trying to use a 12 V DC air compressor which came with a car adapter and integrate it with my Arduino and a relay/transistor. Basically I want to control turning on and off the compressor with an Arduino digital pin but use an external power supply such as a wall-wart for the compressor. I have stripped the wire and removed the car adapter and would like advice on how I can now integrate it into a circuit using a relay or transistor and a wall wart. Would a wall wart provide too much power for the air compressor than a car adapter would? I am unsure of the power rating of the pump so not entirely sure how much current it draws and if a car adapter can be replaced by a wall-wart. I am afraid of damaging the pump.Thanks in advance for any guidance.

Would a wall wart provide too much power for the air compressor than a car adapter would?

It is not possible to "provide too much power". Devices pull power (current). They will pull what they need, up to the limits of the power supply. Power supplies do not push current.

I am unsure of the power rating of the pump so not entirely sure how much current it draws

The size of the wire should give you a clue. A pair of 1/2" diameter copper wires with soldered connections indicates that the pump is going to draw a LOT of current. A pair of 40 gauge wires indicates that it will not.

I usually check the documentation and markings on the piece of equipment in question to determine how much current it draws. This is often marked on the box. As a last resort, check the size of the fuse in the thing that plugs into the cig lighter socket - it draws less than that, else the fuse would blow.

Assuming the power source is a regulated supply, and supplies the right voltage, you never have to worry about the power supply providing "too much power" - the devices connected will use whatever current they want at that voltage. As long as the voltage is appropriate, you're fine - of course, if you connect a 12v power supply to a 5v device, the device is likely to draw more current and quite possibly fail. There exist obsolete wallwarts that are not regulated, and which supply the specified voltage only at the specified load - luckily these are not very common anymore. Measure the voltage without a load, or with a very light load, if you're using a wallwart of questionable provenance to be sure; those wallwarts with the voltage switch on them should be assumed to be unregulated until demonstrated otherwise. In the case of the air compressor, though, it probably doesn't even much care about the exact voltage, as long as it's something that could plausibly come out of a car cigarette lighter port (these are run off the car's battery voltage, which varies from ~12 to ~15, with lots of noise on the line).

I'm going to guess at least 10 Amps. It takes a fair amount of power to inflate a tire. (i.e. 12V x 10 Amps is 120 Watts.) You'll probably smoke the wall wart, but the compressor shouldn't be damaged.

You could measure the current but your average-everyday multimeter won't handle more than a few amps and you might blow the fuse in the meter. You can measure current by measuring the voltage drop across a resistor (Ohm's Law), but you'd probably need to try a variety of high-power, low-resistance resistors in order to get enough voltage-drop to measure, but not so much voltage drop that the pump doesn't run.