What happened when people got sick in colonial times?

The same as we have now. There are a small number of exceptions, such as H1N1 which did not exist back then. But all the other diseases such as cancer, Heart disease, Diabetes, crohns, celiac, Alzheimer's, epilepsy, gastritis, and dozens of others all existed throughout humankind.

However, many diseases were much more common in the 1400 - 1800s than now because of modern vaccines and sanitary methods. Smallpox, for example, has been virtually eradicated today but was very common at the time. Other examples are malaria, cholera, scarlet fever, mumps, measles, and yellow fever. STIs, especially syphilis, were much more difficult to treat and prevent, and were therefore more common. Diseases caused by nutritional deficiency, such as scurvy (vitamin C deficiency) and rickets (vitamin D and calcium deficiency) were also much more common than they are today.

Indeed, Kemtri expanded upon my answer most Excellently.