With regards to schools you have to worry about two things: the FEDERAL law (Gun Free School Zone Act) that prohibits you from carrying a firearm within 1000' feet of a school. You are, however, EXEMPT from this if you have a license from the state in which the school resides, e.g. Delaware CCDW is exempts you from Delaware Schools. PA LTCF exempts you from PA schools, etc.
The section in the Delaware Criminal Code that penalizes carrying a firearm in a school zone actually takes effect ONLY if you are concealing without a license. If you're concealing with a license, or open carrying with a license, that'll cover you with regards to both state and federal law. You could technically open carry without a license and be within Delaware law, but you'd be violating Federal law.
The Post Office (and Federal property) is prohibited under Federal Law... google for that.
A Court House, Prisons, Police Stations are considered "Detention Facilities" and firearms are considered "contraband" and are, thus, verboten in these types of locations.
Any other public property (except state forest, state parks and lands controlled by Fish and Wildlife) should be lawful for you to carry a firearm.
Moosie