Cancer can affect any part of the body, so it can cause almost any symptom known to man, but here are some signs that might make you suspicious

Weight loss - If you've always been trying to drop a few pounds and suddenly you become successful, especially without trying very hard, this may be a sign of cancer. Even if you're not trying to lose weight, if you start noticing that your pants are getting loose, or you weigh regularly and see that your weight is going down - that needs to be investigated. Often, people feel like they are eating enough, but they have actually started feeling "full" quicker than normal, so if they keep track of their diet, they see that the quantity they're taking in has actually decreased. This phenomenon is called "early satiety".
Pain - we all have aches and pains that come and go, but if you notice one that's there on a regular basis, in the same place, especially if you notice it at night (when there are no distractions) at first, then you start noticing it in the daytime as well - that needs to be investigated. Cancer growing in an organ (especially a bone) can cause pain due to destruction of that organ as it grows - indiscriminately eroding through everything around it.
Any new or unusual symptom that persists and becomes more severe with time - a spot on your skin that doesn't heal, a lump or bump that keeps enlarging (especially in the breast, testicle or lymph node area (neck, armpit, groin)), a neurologic symptom like weakness in one arm or one leg, trouble speaking, trouble with vision, early morning headaches (especially if associated with nausea), bleeding from the rectum or with urination, a persistent cough, coughing up blood, trouble urinating or with passing stool, hoarseness, trouble swallowing...