Teaching English abroad is a great way to spend your first year or two out of college and a Classics degree is the perfect preparation. Even though you may not have studied how to teach English, your experience with ancient languages will enable you to address your students' needs with a sensitivity to language studies which other English teachers will not have. The experience can also often lead to other career and life possibilities you never considered. However, even if you already have a long-term career plan, teaching English in a foreign country can provide you with a priceless cultural experience which will certainly enrich your resume.
The Current Global Situation: As a native English speaker with a bachelor's degree, you are in a unique position to find many opportunities teaching English at the elementary, high school, and even college level. Unfortunately, it is no longer very easy for Americans to find jobs teaching English in Europe since priority is given to EU members. This problem is also complicated by the fact that it is becoming more and more common to require various kinds of certification (such as TEFL or TESOL). However, there are still many opportunities throughout Asia.
Getting There: There are many agencies out there which will be happy to help place you in an English teaching position. Some of these can be very helpful, but the truth is that many will charge a lot of money and in the end, place you in a situation which is good for them but bad for you. In general, I recommend not using these kinds of agencies. The best thing to do is to carefully research the country where you are interested in going and then use the internet to contact various institutions. One of the best online resources is Dave's ESL Café. There you will find many job listings in Asia and they will also allow you to post your resume. It is also very important to talk to people who have taught in the same country (or city, if possible).
Which country should I go to? Generally speaking, the more developed a country is, the more difficult it is to get a job without special certification. You will make more money in a developed country relative to U.S. standards, but you will probably make more money in a developing country relative to the local economy. The key is to consider what kind of experience you want. Furthermore, teaching in a developing country does not mean that you have to teach English in a one-room school house in the countryside. You can teach, for example, in Shanghai or Beijing and have more luxuries and a more exciting nightlife than you ever experienced in the U.S.
Here is some advice on individual countries: At one end of the spectrum is China where the job market for English teachers is overflowing and it is easy to find a job even at the college level. Most schools will provide you with an apartment, a return plane ticket, and enough money to have fun and eat out at every meal. Unfortunately, even if you save a large percentage of your money, it will not likely be enough to bother bringing home. At the other end of the spectrum is Japan. You will most likely teach elementary or high school and will probably be assigned a native co-teacher. The money will be great and if you are able to save, it will be worth bringing home. The problem is that the cost of living is so high that it will be difficult to save at all. In between these two extremes are any number of other countries such as Korea and Thailand.
Private Language Centers: In many urban centers throughout Asia, it is becoming more and more common for people to use private language centers to improve their English. The students will often be adults who have already finished college and are working in the business world. The money is great at these kinds of schools, but the jobs are more competitive and often require certification or a graduate degree. One of the largest schools like this which has locations all over the world is The Wall Street Institute.