There are two main forms of entry from Mexico into the United States: over one of many land gateways at the U.S.-Mexico Border, or via airliner.
Land gateways are found on several cities along the border, including those on San Diego, Calexico, El Paso, Eagle Pass, Laredo or Brownsville.
Several air routes fly between Mexico and the United States, including destinations such as Los Angeles, San Francisco, Houston, Dallas-Forth Worth, Miami, Orlando, Newark and Detroit.
by swimming the river or having permission from the government
You only need a valid passport to enter Mexico.
Having a passport and/or Visa is a legal way to enter the U.S. from Mexico.
No
Your birth certificate will not help you. You can enter the US from Mexico with an "enhanced" US drivers license, but only if you are traveling by sea or land, not by air.
As long as you have a US visa with you, you can also enter Mexico.
To gain Arizona and New Mexico from Mexico
Mexico
Yes. Also a US visa.
No. Having a visa for the US doesn't grant you the possibility to enter Mexico; you need the corresponding visa from Mexico.
Canadian, American and European citizens can enter Mexico without additional requirements other than their passports.
Canadian, American and European citizens can enter Mexico without additional requirements other than their passports.
Your passport; if you are not from the US, Canada or the European Union, you also need a visa.Arizona, New Mexico, and Texas border Mexico.