Here are the two organizations who run the schools in Ottawa:
https://ocdsb.ca/
https://www.ocsb.ca/
Note that Catholic schools are "free" and public as well. You may be able to get your child into Catholic schools even if you're not Catholic. But you can't complain about them being exposed to Catholicism if you make that choice. Though Catholic schools in Ontario aren't particularly "fire and brimstone".
Those two school boards above cover both the inner city and the vast majority of Ottawa's suburban areas (Orleans, Barrhaven, Kanata, Stittsville, Riverside South, etc.)
Funding is strictly per pupil so there is no real disparity between the resources available to schools in poorer parts of town vs. richer ones.
Of course, the more affluent the neighbourhood, the better your chances are of having most classmates coming from households that take education seriously - as per the usual story.
Ottawa also has networks of free schools that teach in French (public and Catholic).
Here are their websites:
https://cepeo.on.ca/
https://www.ecolecatholique.ca/
Children generally have to have at least some knowledge of French in order to be admitted to these schools.