Alright, I didn't catch your location before I hit reply, but I'm assuming you're in the US. In the US Archaeology is considered to be one of the four fields of Anthropology, along with Linguistics, Cultural Anthropology, and Biological or Physical (sort of an outdated term) Anthropology. That is why Archaeology is housed in the Anthropology Dept. To become an Archaeologist you need to receive your BA or BS in Anthropology. Most departments let their students focus on one of the four subfields and take specific courses that lead towards that part of the field. eg. I majored in Anthropology as an undergrad, focusing in Archaeology. I had to take archaeological methods and practical courses, along with statistics, chemistry, all along with my required cultural and biological courses that were expected of every Anth major - which were mainly introductory courses into each subfield, then upper division courses in specific theories, methods, eg. Kinship Theory, Biological Principles, Origins of Modern Humans, and also ethnographic or archaeological courses, like Native Peoples of North America, Eastern Woodlands Archaeology, etc.
Really, the required course load depends on the school you attend. My undergrad institution just revamped their curriculum, and courses like computer programming are no longer required, while some history courses now are (I think, I never looked it over very carefully). But most require the intro courses for most of the subfields, with you being able to choose your upper division courses.
To any aspiring archaeologist - the most important thing you can do when considering an institution is not to just pick a big name school off the list but look at who the faculty members are, do any of their interests align with your own. I'm starting my PhD this fall - not at a big school that is well-known throughout the country, but I will be working with three of the top people in my part of the field.
And then there's the Classics Department (Rome, Greece, etc) - but I don't know much about them. You'll need to find a Classicist for that.
Hope this helps.