I started on trumpet, then switched to cornet the next year, due to financial concerns. The cornet was cheaper, and the band system I came into DID NOT allow trumpets. The sound model for the group was American style cornet, not BBB style. We all used the Bach 7C or Conn 4 until high school, no exceptions. In high school we were allowed to play whatever mouthpiece got the job done, and this is where I learned you can have multiple mouthpieces and a single instrument, and create different tones. I used a Bach 5A for concert band, and a 10 1/2C for marching band.
For the OP, if I were a teacher and had a choice in the American educational system, I would opt for cornets. They are a lot cheaper here, and so few band systems use them the odds of a student getting more bang for the buck goes up. In a perfect world, I'm want trumpets for the field and pep band, cornets in concert band with the trumpet parts covered by a C cup mouthpiece in a cornet.