Are there any general guidelines about when to start health screenings such as mammograms and colonoscopies? What if I have a family history?

Yes, especially if there is a family history. Women aged 50 to 74 should have screening mammography every one to two years. Some women younger than 50 will need to be screened depending on their family history and some will need to be screened more often.
Women should have PAP smears beginning at age 20 or within three years of becoming sexually active. Routine screening is typically every three years for those 21 to 65 years old. Some providers may shorten or extend the screening interval based on history and risk factors.
Men need to have an annual PSA (prostate specific antigen) beginning at age 45 to 50. Both men and women should have screening colonoscopy beginning at age 50, depending on personal risk factors and family history; they should also have fecal occult blood testing each year.