What the gurgling is actually telling you
That gurgle is the sound of air fighting water in your pipes. Drain water needs air behind it to flow smoothly, the same way you punch a second hole in a juice can so it pours.
When your drain can't get that air from its normal source (the roof vent), it grabs air through the closest weak point: the water sitting in your P-trap. The bubbling up through that water is the noise.
So gurgling means one of two things. Either something is restricting the drain line so water and air get squeezed together, or the vent that's supposed to feed air is blocked.
The good news: on a kitchen sink specifically, this is often a weekend fix for under $90. The trick is reading the clues before you start swinging tools.
The usual suspects, in order
Most kitchen gurgles trace back to grease. Kitchen drains cop the worst of it: warm fat goes down as a liquid, cools in the pipe, and builds a ring like plaque in an artery.
That ring gets narrow enough that draining water can't pass cleanly, so it pulls air with it. You'll usually notice the sink draining a hair slower too.
The second big cause is the vent. If your kitchen ties into a roof vent, anything blocking that pipe (leaves, ice, a nest) forces the drain to breathe through your trap instead.
The third one catches people off guard: the air admittance valve under the sink. Lots of islands and remodels since the early 2000s use these instead of running a vent up through the roof. They're springs and rubber, and springs and rubber wear out.
The one you don't want is the main line. If your kitchen gurgles when you flush a toilet on the other side of the house, the problem is downstream where everything shares a pipe. That's a plumber's call, and sooner is cheaper than later.