My kitchen sink drain has a complete clog. I have a dual tub sink with a disposal on the left and the dishwasher drain on the right. The water backs up into the sink regardless of which tub I run the water into. It also backs up into both sinks when I run the dishwasher. This tells me the disposal isn't the problem. All of the other drains in the house are fine. I tried Draino, that didn't touch it. I took apart the P-trap, and there was nothing blocking it there. With the P-trap still off, I shoved about 8 feet of snake in the outlet pipe. I couldn't get it to go any further than that, so either I've hit the blockage or I've hit a bend in the pipe that I can't get the snake around. I put the P-trap back on and tried stuffing a rag into one drain and plunging the other one. That didn't work, because I couldn't get an airtight seal. I started tightening the nuts to see if that would create a bigger seal, and busted the outlet nut on my P-trap. Unfortunately, my P-trap is non-standard. The outlet pipe going back into the wall is 2", not 1 1/2". I haven't been able to find a P-trap anywhere that has 1 1/2" inlet and 2" outlet.

So now I guess I have to cut my outlet pipe, put on a reducer, then put on a standard 1 1/2" to 1 1/2" P-trap, right?

And once I get that done, what do I do about the clog? I don't have a basement, my house sits on a slab. Is there supposed to be a vent somewhere that I could check? It would have to be a vent for just the kitchen sink drain, since everything else is working.

You will find the blockage in the lateral ,(horizontal) drain line in the wall. Either your snake is too small or you missed the clog but you'll have to go back and do it again. Send your snake in the pipe coming out of the wall. You will hit a elbo about 6" in but crank and work your way around it. About 6 morefeet and you'll hit the drainage tee in the stack. If that don't get ityou have a blockage downstream from the drainage tee and you'll have to rent a sewer machine and snake out from the kitchen roof vent. Let your cable drop down to the base of the stack and put out about 20' more. Afterwards flush out thye line with a garden hose.
As for the 2" compression nut that you broke, replace it and pick up a 2 x 1 1/2" nylon compression washer to connect your 1 1/2" trap to the 2" stubout. Good luck, Tom