Can I get certified to teach primary school and high school? (Ontario, Canada)?

One possibility is to take additional Teachers' Qualification Courses such as Primary Specialist or a Senior's Qualification in a specific area. Unless the regulations have changed the Senior Qualification has pre- requisites. Since the qualification courses are under the auspices of the Ministry of Education go to their website for further information. Another source of information is the Ontario College of Teachers (OCT.ca). You can also contact the EQAO for information as well as the teachers' federations. As an example for a Senior Qualification in History you would need to have taken some university history courses. When I graduated from Toronto Teachers' College I was qualified in Primary, Junior, Intermediate. I began my career at the elementary level but finished at the secondary level. Along the way I picked up several Senior Qualifications + Specialist Certificates.
Do keep in mind that while the Board of Education and the individual principal will take your preferences into account the reality is that you will be placed where there is a need. In general the individual school asks for the teacher's preferences for the following year. So if you are teaching a grade that you don't particularly want that is when you make your request. Another possibility to request a transfer at the end of the school year. Boards usually announce upcoming vacancies in their news bulletins which is distributed to all schools and it is your responsibility to read them