There are typically two bowls in kitchen sinks. That complicates things when you need to plunge a kitchen sink. Plungers work via pressure, but an extra pipe and another sink actually diffuse the pressure a plunger normally builds up in a single-bowl fixture. Still, plunging a dual-bowl sink can get rid of some clogs, although tougher ones won't be as affected.
Plungers require pressure build-up in order to function. The problem in a dual sink is that you can plunge one side only to have the pressure go out the alternate drain, leaving the clog in place and untouched. It's useful to have a second set of hands here, as someone can hold the stopper in position in the other drain. If no pressure can escape it, your plunging on the first drain has better odds of success.
Be sure that the plunger is covered in water. Also be sure the stopper in the other drain is secure, probably by someone holding it. Hopefully, plunging works, as snaking kitchen sinks is far more work and can also risk a huge mess. Still, if you want to try it, sink augers of twenty five to fifty inches in length can be had for twenty dollars or less.
Some augers feature a crank, whereas others just sport a movable handle which makes turning the snake possible. An electric snake is rarely necessary for sink drain issues, unless of course you have one of the kinds that can be attached to a power drill. If that's available, try it!
The wall is where you want to run in your snake. Using a pair of adjustable pliers, remove the nut of the outlet leading into the wall.
You're going to need multiple turns and constant pressure to get the snake around bends in the pipe. The twenty five foot augers are usually sufficient for most sink drains, but on some occasions, you have to go deeper. Expect to encounter resistance when you reach the clog. If water is still in the pipe, it will make audible noise if and when the clog is broken.
Run your snake forward and backward until it has freedom of movement. Once done, don't just yank it out, otherwise you'll spray dirty water and slop everywhere. Reassemble the drains you worked on and test out your sink. With luck, you've taken care of the problem yourself relatively quickly and cheaply and can go back to enjoying your kitchen as a source of good meals instead of headaches and work.