Power a MOC3010 with an arduino
Sadly your MOC3010 might not be up to the job. The datasheet shows that the Triac current is an absolute maximum of 100 mA RMS. Most basic doorbells can draw much more than this on the 24 V AC side.
If you have a multimeter, you can test the current drawn by the doorbell, and if it's less than 100 mA by a good margin (say only 50-60 mA AC) then it may work. 

The maximum LED Diode drive in the datasheet is 15 mA which is in range of the Arduino pin drive (20 mA) so the series R would be 270 Ohm for a 5 V Arduino (and 139 Ohm for a 3.3 V Arduino).

The project hangs on the 24 V AC current drawn by your bell.

You will notice in the datasheet that the MOC 3010 is usually used to drive a more powerful Triac, you could certainly do that to handle a higher current on the 24 V AC side. 

Now I hesitate to mention this as this means you working with much higher voltages. You could switch the 110-120 VAC side of the doorbell transformer to achieve the same result. The MOC3010 would be able to handle this and the AC current likely is less than 50 mA; and even allowing for any surge the Triac has acceptable ratings.
If you are confident doing this then that may be a solution.....if you lack the confidence then I'd suggest you buy a Arduino compatible SSR or relay board from the likes of SainSmart. Here's a link to Amazon for a 2-channel relay, but there are endless devices SSR and relay modules like this.
Much safer......