You can earn skill points by going to school, you're job (you can get a job when your tamagotchi is a Adult), feeding it certain foods, and by playing games. Well that's about it.
There are three teachers which match up with each skill point. Mr. Turtlepedia is the Intellectual teacher. Ms. Flower is the Arts/Fashion teacher. Mr. Canvas is the Social/Kindness points. To pick your teacher, just press A (first button) to scroll through the teachers. Press the B button (middle button) to select the teacher you desire. Make sure that you don't leave your screen on a teacher too long - you will end up enrolling in that teacher's class