One immediate thing to keep in mind: there are more options to graduate school than simply the Ph.D. Depending on what your goals are, you may find that only one or two years of graduate work (e.g., for a Masters degree) will give you what you need.
Graduate school is a commitment: make no doubt about it. The workload can be intense. Pursuit of the Ph.D., for example, will take you several, usually at least five or six, years. Making big money is not in the cards. On the more positive side, you spend your time studying a subject that means a great deal to you, and have the opportunity to work and talk with other people who share the same passion.
Most of the specifics of “how to proceed” will depend on which particular disciplinary track you wish to follow: classical versus Egyptian versus Near Eastern archaeology, and so on. Each track has different requirements and expectations, and every individual who applies possesses different strengths and weaknesses. As a result, there is no straightforward, “cookie cutter” model for the process of identifying good schools and applying to them. Students are strongly encouraged to talk to their concentration advisor or other mentors early and often about their graduate school ambitions. In addition, careful research (for example, using the web resources listed below) will help you make intelligent, and efficient, decisions.