In Baltimore, children under six can ride free with an adult who pays the full fare. They have a special student program for K-12 students which offers a reduced fare if the student signs up for the program. Otherwise, chilren over the age of 6 have to pay full fare.
Back in the 1960s, I was able to ride the Paoli Local on the PRR at half fare until I was 12. I didn't need to have an adult accompany me. When I would ride the Redd Arrow Lines and the PTC Market-Frankford Line, I had to pay full fare. When I was in high school, the School District of Philadelphia didn't rely on school buses, but it had a deal with PTC (later SEPTA) where we could buy a pack of 10 "school tokens" for 85 cents. (The regular fare was 25 cents.) The school tokens included free transfers, so I could ride all over the city in the afternoon is I had wanted to. If I wanted to ride on the weekends, I had to pay the full adult fare.
If I rode the Penn Central or Reading commuter trains, or took joyrides on the NEC, I had to pay full fare. However, during the Christmas and Easter school breaks, Penn Central offered a special deal for kids 15 or under on what they called "ladies' days" (Tuesday and Thursday, I believe.) Adult women could ride to New York on a round trip for half fare, and during the school breaks, kids 15 and under could ride for half of that. I did do that once with a friend when I was in 10th grade. We went up to New York for the day, rode the subways and went to the Museum of Natural History to look at the minerals and dinosaur skeletons.