How are cancer cells able to get nutrients away from your healthy cells?

well this is an interesting question and i would say it may depend on what type of cancer cells you are growing and what type of "normal" cells your growing. One possibility is that cell fusion events may occur between your cancer cells and normal cells, thus creating a few options 1 - making the normal cell cancerous, 2 - making the cancer cell that fused with the normal cell not cancerous anymore. 3 - either way the fused cell will have a different genotype and hence be a different cell.