My sink is backed up, meaning the water won't drain. I have tried to turn on the garbage disposal, but when I do, water just kind of bubbles out of the disposal drain. I've just run the dishwasher and the water appears kind of soapy and is warm to the touch. Help?

As a rule most sinks clog up in the horizontal drain in the wall. Try this.
Dip and sponge the water out of the sink and remove the "J" bend from the trap. Now run your snake up into the wall,(see image) for 7 feet. This should get the clog. If not then click back and we'll go to plan "B", Good luck, Tom

Hi max. Is this a one or two compartment sink? If two, is water backing up into the other side, or is it just the garbage disposal side that is giving you issues? If just the garbage disposal side, plunge the heck out of that thing and you should be all set. Sounds like the garbage disposal is simply a bit plugged up. Your dishwasher drains into the garbage disposal, so that is why you are seeing the waste from the dishwasher.