I already applied to get my US passport but I don't think it will be ready on time, I'm also a Mexican citizen, so, if I'm planning to fly from San Diego to Mexico City, can I do it with a Mexican passport? I also have a current green card that would probably work for the way back. I need to know as soon as possible, otherwise, I'll need to cancel my flight.
Whatever you do, DO NOT ATTEMPT TO ENTER ON A GREEN CARD if you are a U.S. citizen. When you become naturalized, your green card is invalidated (you are supposed to surrender it as soon as you find it), so doing so can count as entry using an illegal document, and can lead to prosecution.
To enter the U.S. from Mexico, you will need proof of P.R. status (or citizenship, in your case). If you can show them your U.S. passport on your way back, then you’ll be allowed back in. If you can arrange for this to work (e.g. have someone mail you the U.S. passport), then you should be able to get back in.
It is technically illegal to leave the U.S. on a passport different from the U.S. passport. Mexico will let you in if you have their passport, but there might be problems on the way back. Again, I am not aware of any penalty for doing so; however, you might have some explaining to do.