How many PhD programs should I apply to to be sure of getting in somewhere?

The answer depends on the field of study. Ask several of your current professors in the field.

Personally, I would apply to at least ten graduate or professional schools, no matter what the field of study. For most fields, competition is that intense.

For psychology and other very competitive fields you may want to apply to over twenty programs.

It really depends whether you barely qualify or whether you are doing research that is seen as valuable and whether you did a fellowship in high school.

My daughter gets her grad degree this spring ans has applied for no PhD programs. Instead she has had 9 schools contact her and try to recruit her. Including schools like Berkeley, MIT and Texas Tech. She did her first thesis as an undergrad. She discovered a flaw in how funds are allocated for emergency services in remote areas and has found that this is killing several people a year. She will be researching to calculate how many and how to fix this problem.

If you have a middling GPA in say the 3.5 range and your research has not gotten the attention of someone as valuable or you did not do a fellowship then you should apply to at least 6.

Depends on your GPA. If it is below a certain minimum — 3.5~3.0 in many fields — you should not apply at all.