For an indoor track, you should ask the coach or athletic director about permission and times when it is open to the public. There are lots of liability trespassing and issues that are easier to enforce with an indoor facility. Some schools will have an athletic club style membership plan that allows the public to use their facilities (swimming pool, tennis courts, fitness centers, etc.) when they don't conflict with schedules school activities (PE classes, team practices, etc.).
For an outdoor track, there are usually signs indicating when they are open for public use and/or reserved for team practices. It's easier to run on an outdoor track without asking for permission, and if a guard or coach tells you it is closed, then you could ask when it's open to the public.
If you want to use a field event facility (e.g., pole-vault pit), then fewer people will have a "don't ask, don't tell" attitude about this versus running due to the safety issues. You may need to join a youth/master's track club that provides liability insurance and has formal access to a school's facilities.