When you write that you want to teach English abroad, do you mean English lit and composition, or do you mean English as a second/foreign language? The preparation for those two are different and lead to somewhat different career options.
To teach English lit and composition abroad in private or department of defense secondary schools would require a US teaching certificate and at least two years of teaching experience in a secondary school here. Recruiters rarely (as in almost never) hire fresh college grads for those jobs.
To teach English as a second/foreign language abroad might not require anything at all beyond the willingness/desire to do so - this often is country-specific. For a better teaching experience, and a better pay scale, a college degree in anything and a CELTA certificate that takes about a month to complete will do. Many language schools will hire CELTA program graduates. For the best options, and best future employment options on return to the US, you want to complete a bachelor or master degree program in TESOL that leads to state teacher certification.