At the same time, nothing is more rewarding than the study of the classical world, if you have an all-consuming passion for it. I recommend to all my students that they test their passion by taking a year or two off before trying graduate school, to see whether they can (or can't) live without academia. If nothing else satisfies, then all the warnings anyone gives won't make a difference. But that's something that only you can decide for yourself.
For those who are thinking about graduate school in my field, the following information is intended to give a sense of what preparation for graduate school involves, what graduate study itself may be like, and some further resources about both particular graduate programs and about the field of ancient history in general. I make no claims to completeness or comprehensiveness, and I urge folks thinking about graduate school to seek counsel widely and often from people who have been there.
Preparing for graduate school.
Greek and Roman history lies at the intersection of two fields: Classics and History. In general, there are two approaches to undergraduate preparation. Major in Classics, and take as much History as you can, or major in History and take as much Classics as you can. Neither approach is ideal, and both have special benefits. What you do may depend more on your own tastes than anything else, but I would urge everyone thinking about graduate school to be sure to take archaeology and one general Classical Civilization course in Greece and one in Rome. Excavation experience as provided by Caesarea is extremely useful; it would also be very helpful to have participated in a surface survey as a volunteer.
Languages. Nothing is more important than the study of languages. Ideally, an applicant to graduate school in Greco-Roman history will present four years each of Greek and Latin. In addition, and as a minimum, you will need to be able to read comfortably in German, French, and Italian. Depending on your particular field of concentration, you may need any of a number of additional modern European or Middle Eastern languages, or further ancient languages, such as Old Persian or Syriac or Hebrew. The more you can get as an undergraduate, the better off you'll be in graduate school. As a practical matter, of course, it is very hard to do all this as an undergraduate; I would therefore recommend that you concentrate on Greek and Latin and then take an additional modern language, probably one you did not have in high school. (If you took French in high school, take two years of German or Italian here.) If you want a specific recommendation, I'd encourage German. German is the hardest of the common modern languages and there is an incredibly rich literature in German on the ancient world, yet too many ancient historians feel weak in German.
Breadth always helps. I recommend that you take everything you can outside your major. In particular, anthropology, economics, philosophy, and statistics are extremely useful. The last is especially recommended as too many historians are scared of statistics and don't understand statistical arguments when they read them.
Read. Read everything you can, especially journal literature. Familiarize yourself with the bibliographic resources for the study of antiquity, particularly L'annee philologique, the greatest source for new publications in all areas of ancient study. Use your reading in the papers you write; don't be afraid to do more and harder work than your professors demand. Summers are a great time to do additional reading; make up a reading list for yourself every summer (with help of faculty if you like), and then read it.
Travel. If you can get to the Mediterranean, go. Whether for a week or a summer, any time spent there will repay its costs a hundredfold. Nothing can replace an actual personal knowledge of the topography and climate of the lands in which occurred the history you want to study.
Write a senior thesis. There is no better test of your vocation for graduate school than writing a serious senior research thesis. A very good senior thesis of (say) 100 pages is practically the equivalent of a typical MA thesis, and indeed may serve as the basis for MA work (if your advisor allows). In any case, the ability to sustain an argument over a long stretch, relying on primary sources to produce an original contrbution, is the best predictor of success in graduate school -- assuming that after a year of research and writing, you haven't decided that you hate the whole business and just want to get a job!
What's graduate school likely to be like?
When I was an undergraduate, I shared an apartment one semester with a graduate student in physics. He described the difference in work between undergraduate and graduate programs as "a step function." That is to say, graduate school entails far more work than even the toughest undergraduate program you can imagine.
There's considerable variation from program to program, but generally here's what you can expect. The first few years focus on coursework. You'll be taking Greek and Latin courses every semester and seminars in Greek and Roman history. Graduate seminars are usually very different from undergraduate seminars. Threre's lots more reading (on the order of hundreds of pages per week), often including material not in English. You're expected to be much more independent: often, you will be required to compile bibliographies with very little guidance. Most seminar sessions may be devoted to presentations by students that may last an hour or more; you will have to become an instant expert on your topic, knowing all the primary sources and the principle secondary literature.
Treatment of the Master's degree varies considerably. Some programs do not admit students to the Ph.D. program but require students to complete a master's first and then apply for the Ph.D; some simply grant an MA en passant, as it were; others have dispensed with the MA altogether. For myself, I think it's useful to write an MA thesis as practice for the Ph.D.; if you can't complete the MA thesis, or hate it, you may be getting a strong warning that continuing on for the Ph.D. is not for you.
At some point, students are required to pass Comprehensive or Qualifying Exams. These exams may serve as a gateway to the dissertation, or may be required after the first or second year of the graduate program as a way of weeding folks out early. These exams typically consist of essay questions and may last anywhere from two days to a week. The exams are designed to test your comprehensive understanding of the history of the Greco-Roman world, and perhaps also of ancillary disciplines (archaeology, literature, and so on). You will probably be required also to pass exams in Greek and Latin and in two modern languages.
The last stage is the research and writing of a Ph.D. dissertation. This task can occupy anywhere from one to six or seven years, depending on your topic and personal circumstances. The final product is a manuscript of (say) 150-600 pages which makes an original contribution to the field.
I should say a word here about your Ph.D. advisor. There will be no greater influence in your academic and intellectual development than your Ph.D. advisor, and it is important to have this in mind even as you begin to apply to graduate school. As you read about (and meet) the faculty in the programs you are considering, always ask yourself: how would I feel about working under him/her if I decide to write a thesis in his/her field? Poor fit between advisor and thesis writer is probably one of the most frequent causes of problems at the Ph.D. stage.
Choosing a graduate program.
There are many considerations to choosing a graduate program, but I would especially recommend thinking about four areas: faculty, fields, funding, and placement.
Faculty. Nothing is more important than the faculty with whom you will work. Certainly, you'll want to be studying under the best people you can, and that should be judged in part at least by reputation of the faculty and their publication records. (Reputation of faculty and reputation of the program are two different things. The world expert on some area may well be teaching in a second-tier institution, while a first-rate program may be riding on a reputation established long ago and which the current faculty cannot actually sustain.) Ask for publication records, and check the bibliographic resources that you have become familiar with through your undergraduate work. Make sure faculty are up-to-date in their familiarity with their fields.
At the same time, the greatest scholar in the world is useless as a teacher or mentor if s/he won't meet with you or read your papers. When you visit graduate programs, test the accessibility of the faculty. Is it easy to get appointments? Are they often around? Ask current graduate students you meet.
Fields. Even the best programs are going to be weak somewhere. Before you start, you should have an idea of where your interests lie (Archaic political history? colonization? Roman expansion in Italy?) and make sure that the programs you consider are strong in those fields. This means not only that they have world-class faculty in the fields, but also access to other resources in the fields, like archaeological excavations, good library holdings, computer resources, and so on. Bear in mind that your area of interest may well change over the course of your study, so be sure that other areas of apparently peripheral interest are also supported.
Funding. It is in my view crucial that the program you choose have good funding sources, both internal and external, to support graduate studies, and that they show interest in funding you. Graduate school is unconscionably expensive, and I would never recommend anyone to go expecting to finance the costs entirely through parental gifts, work, or borrowing. If your favorite program cannot provide assurances about future funding -- and best of course to start with a fellowship -- then you should seriously consider looking elsewhere.
Placement. The best program in the world is next to useless if it does not seek aggressively to place its graduates. You want a well-established, efficient graduate placement service; faculty committed to seeking positions proactively for their students; and an established record of outstanding placement for recent graduates. Beware of the programs that identify a student or two as "stars" who then receive all the attention, while the rest of the crowd is left to fend for itself.
Most important: visit campuses, and talk to current graduate students. They'll be your best sources for the inside story.
Some graduate programs.
For a sense of what jobs there are, skim the American Philological Association publication Positions for Classicists.
Here are some universities with graduate programs in Greco-Roman history: