In the US at least, "grade school" is the same as "elementary school" - you attend there between the ages of around 5 and 11. After that is "middle school" or "junior high," then "high school." Graduate from high school and you've completed public education.

After that is "college," what a lot of the world calls "uni." It's generally not free. First two years are basically an extension of high school with a few degree-specific classes. You can get an Associate's degree from that. The third and fourth years are almost all degree-specific classes. Finish that, and you have your Bachelor's degree.

College up to and including your Bachelor's degree is called "undergraduate." Most grants, scholarships, and other financial aid ends here.

The next stage is graduate school, where you earn your Master's degree. After that is your postgraduate, where you work towards your PhD.

Trade school in the US (also called vo-tech, or vocational-technical school) exists as well. Some people take it at the same time as high school, others take it instead of college. Sometimes you get a degree, but often it's a certification or license. Trade schools usually aren't free but often there are programs you can sign up for that pay for it.