The cells that carry nutrients from food to the rest of the cells in the body are primarily the red blood cells, particularly through the mechanisms of the circulatory system. However, it's important to note that the nutrients from food are absorbed by the intestinal cells (enterocytes) in the small intestine and then transported into the bloodstream. From there, the nutrients are carried by the blood (and thus by red blood cells) to various tissues and cells throughout the body for use in metabolic processes.
In summary, while red blood cells are crucial for transporting oxygen and, indirectly, nutrients, it's the plasma in the blood that carries the nutrients absorbed after digestion to cells throughout the body.