Most classical degrees are of the following kinds.
The names vary from one institution to another, but basically you have to ask yourself these questions:
If you want to take Greek and Latin seriously, look for degrees involving 'papers', 'courses', or 'modules' in the languages. These degrees are typically called BA Classics, or BA Greek, or BA Latin. At Oxford, classics is called Literae Humaniores, for which (as in BA Classics elsewhere) you can study a mixture of different subjects such as literature, history, archaeology, and philosophy. There are also joint degrees such as BA Latin & French.
If you don't want a degree that's heavy on language, look for degrees with titles like BA Classical Studies, BA Classical Civilization, or—if history is your favourite—BA Ancient History. These subjects are also available in many joint degree combinations, such as Ancient History & Archaeology, Ancient & Modern History, etc.
If you want to avoid doing Greek and Latin altogether, you will still find that most of the degrees called Classical Civilization, Classical Studies, and Ancient History will allow you to do this.
There are also in some universities degrees like Combined Studies which will give you greater flexibility. You could, for example, combine Latin with Bronze Archaeology and French. Just to give some examples: in 2001, there were degrees available in African Studies with Archaeology & Ancient History; in Archaeology & Ancient History with East Mediterranean History; and in Archaeology & Ancient History with Modern Greek Studies! If you do this, of course, you won't get to do so much so much of any particular bit of the subject.