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.

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.