What are the core symptoms of cancer?

Varies with type. Cancer is many different illnesses and the core symptoms vary with the organ affected and the type of cancer. The following are some of the core symptoms: lump in an organ, weight loss, blood in stool, urine or sputum, cough, tiredness, vague abdominal symptoms including pain, unexplained infections, persistent headache, change in bowel habits, irregular menses etc.
Almost anything. Any symptom can have a whole range of causes. These causes generally fall into certain categories. Only one of these categories is cancer.