Do not attempt. A smoke detector is a life safety device, designed by people skilled in that art, tested exhaustively, and listed for its application. When you say,
"I'm fairly new to electronics ..."
you're telling us that you're absolutely not qualified for this work. There's nothing about an Arduino that suggests that it's intended for a life safety application, either. If you inadvertently disable the detector, and a fire occurs, occupants won't be alerted, and that's a very bad thing.
Having said that, there is a way to trigger the Arduino that's consistent with the use of the detector. Mount your new gizmo as close to the smoke detector as the manufacturer's instructions allow, use an electret microphone to listen for the sounder, and program the Arduino to detect the oscillations. That's likely to be challenging for a guy that's fairly new to electronics - it's a character-building experience.
But, don't get into the smoke detector. Don't try to steal power from it, don't connect to its input power circuit or its signalling circuit, if it has one.
If you just can't resist doing something inside a smoke detector, then buy another one, fiddle with it, and mount it in addition to the existing detector(s), if there are any, or the one(s) the codes require, if there aren't any. Then, paint it black, or otherwise mark it so that no one will mistake it for a smoke detector.