According to what I have read, it seems to be that about 8% to 15% of people who have had a melanoma will develop another within 3 years. I read an article on
MEDEM about this. So, your chances are pretty good that you won't develop another as long as you take good care of your skin. It can also depend on your total sun exposure over your whole life. Don't worry about it too much. Melanoma can usually be treated successfully if caught early. I would advise that you do a self-exam for skin cancer once a month. Just look all over your body when you get out of the shower and look at hard-to-see parts with a mirror. Get to know what moles and freckles you have, so that you will notice if something changes. Ask your doc how often you should have a skin exam at the doctor. It seems most people should do this at least every six months if they have a history of skin cancer.
As far as your chances of developing other kinds of cancer, that depends on your genes, habits, and physical condition. Exercise, proper diet, and regular physicals will help prevent cancer, or at least catch it in the early stages. If you have a family history of certain kinds of cancer, or if you do not know your family history, ask your doctor about screening tests appropriate for your age.
Don't worry about it too much! Stress isn't good for you. You had cancer and beat it! My mom had a melanoma 7 years ago, and she hasn't had anymore since. If you feel that your doctor isn't paying enough attention to your concerns, ask for a second opinion. I hope this helps! Congrats on beating your cancer!:)