I am a citizen of Mexico (where I was born) and of the United States. Is there any way that I could travel without my passport either by land or air? Only documents I have on me are my state ID and my social security card. Are there any other options I should consider?
Without reliable proof of US citizenship, you are liable to be refused entry to the US. As you were born in Mexico, such proof would be a consular record of birth abroad or a naturalization or citizenship certificate.
As I mentioned in a comment, a social security card is virtually worthless on its own as proof of citizenship, the main reason being that anyone authorized to work in the United States, citizen or otherwise, gets a Social Security number.
It might be possible for a CBP officer to look up records of your naturalization (or birth abroad to a US citizen parent, as the case may be), but I would not count on it. In any event, an airline won't do that. This approach would only be suitable for a land crossing attempt.
If you find yourself in Mexico without a passport and wish to travel to the United States, your best chance of success will be to go to a US consulate and apply there for a passport or some emergency travel document. This could of course be time consuming and is likely to be expensive, so, depending on where you are and on how much money you can spend, you might try your luck first at a border crossing.
As described in various other answers on this site, while you are technically supposed to only enter the US with your passport you cannot be refused entry to the USA if you are a US citizen. Airlines will not help you without a passport, but if you show up at a land border you may have luck.
The trick is proving you are a US citizen.
In this case with and ID and social security card it seems that should be enough to 1) prove you are who you say you are (ID) and 2) prove that you are a citizen (ss card) gaining entry to the US.
However, be prepared to have some explaining to do and leave plenty of extra time at the border.
Visiting or traveling to any country without legal documents like Passport is a crime, Passport is not only the authenticating document but also your verification and safety while traveling. As Mexico is considered one of the most exciting tourist places, one must respect the citizenship of such country.