What are some carcinogens that we encounter on a daily basis that we should try to avoid? or that we should just try not to have too much exposure?

Riskiest: tobacco: There are too many to list, but some big ones that you can choose to avoid are tobacco products and UV light. There may be some carcinogens in food additives, but these are small and largely effectively regulated by the FDA. Industrial toxins and carcinogens can also be a concern, but this is one of the ways the EPA helps us. Your own advice is best: limited exposure vs. total avoidance:)

DNA damage may occur: ... from a variety of natural sources in the environment. Things like cosmic rays which ionize water molecules can not easily be avoided. Things that are avoidable include cigarette smoke, air pollution, asbestos, and indoor radon gas. Some chronic infections can cause cancer if untreated, including HIV, and hepatitis B or C.

Tobacco, ultraviolet: These are the two that you can reasonably avoid. Don't smoke, and use sunscreen. Alcohol is a relatively minor risk if used in excess. Protected / monogamous sex shield you from much more, but HPV is a real cancer causer. Most "pop" claims about this-or-that causing cancer are junk-science and you simply can't avoid most of them anyway. Background radioactivity is unavoidable.