As a private studio teacher there are a few basic concepts students must learn that are of the most extreme importance. These are tone, intonation, and mobility of the fingers.
The first is that of tone production. A cellist's "voice" is the sound that comes from the instrument. A player can have wonderful intonation and musical expression, but if no one can hear them, what is the point. Music performance is about sharing. If a young performer is playing in a medium-large sized hall in an important performance and mom and dad are in the balcony, a small sound will be hard to hear. "Johnny looked great, but we couldn't hear him." Learning the subtleties and nuances of controlling the bow is crucial to this development. I train the students to use arm weight to pull a fuller sound out of the instrument. Slow "spun" tone exercises are used for control.
The second important concept is intonation. In addition to proper placement of the fingers on the string and form of the hand, learning to match pitch is just as valuable. I believe students can increase their ability to hear proper pitch through listening to recordings, playing with a fixed pitch instrument and singing in the lesson or in a choir.
Mobility of the fingers is basically how fast the student can lift a finger and place it back down on the fingerboard. The use of Cossmann finger exercises is perfect for this development. Both the single string speed exercise and the double stop finger trills work great. Speed work on scales is also used along with other strength building exercises.