I tend to teach by having students work on primarily Classical music, even if they are interested in other styles. I can teach cello for pop or rock playing as well, however, and I also teach improvisational skills. These could form a part of the student's education even if they are primarily studying Classical music, but it depends on the student's personal interests.  

I feel that learning Classical works is an important part of exploring cello technique, but in the end the student's interests are most important, such as a goal of getting into a music conservatory or joining a band.

Intermediate and Advanced students can expect to work on a Sonata, a Concerto, an Etude, and a movement of a Bach Cello Suite at any given point in time, though this is flexible for Intermediate students.   
Beginning students will work on short pieces and work their way into simple sonatas and accompanied works, along with etudes.   

I prefer to use the method book by Feulliard called "La Methode du  Jeune Violoncelliste" for young students, and for intermediate students I add other etudes, such as Dotzauer etudes. Advanced students would work on the Popper etudes, or some of the advanced Dotzauer etudes. Students interested in non-classical music will have the opportunity to play pop, rock, folk, blues, and any other music styles they are interested in.