das Lehramt
teaching profession
The profession or work of being a teacher at a school.
Nach dem Referendariat trat sie direkt ins Lehramt ein.
After her teacher training placement, she entered the teaching profession directly.

teaching profession / teacher training
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teaching profession
The profession or work of being a teacher at a school.
Nach dem Referendariat trat sie direkt ins Lehramt ein.
After her teacher training placement, she entered the teaching profession directly.

The word 'Lehramt' (teaching profession / teacher training) is made up of 'Lehr-' (from the verb 'lehren' (to teach)) and 'Amt' (office, position, public duty). So it literally means 'teaching duty' or 'teaching position'.
Sie strebt eine Karriere im Lehramt an.
She is aiming for a career in the teaching profession.
The combination 'auf Lehramt studieren' (to study teaching) is a fixed expression. The preposition 'auf' is fixed here and cannot simply be exchanged. This is how you express that someone is completing a degree with the goal of becoming a teacher.
Er ist nach Berlin gezogen, um auf Lehramt zu studieren.
He moved to Berlin to study teaching.
When someone 'auf Lehramt studiert' (studies teaching), you can specify the type of school more precisely with the preposition 'für' (for). Typical additions are 'für Grundschulen' (for primary schools), 'für Gymnasien' (for grammar schools / academic secondary schools), or 'für Berufsschulen' (for vocational schools).
Sie studiert auf Lehramt für Gymnasien mit den Fächern Deutsch und Geschichte.
She is studying teaching for grammar schools, with German and history as her subjects.
When referring to the profession, 'Lehramt' (teaching profession) is often used with verbs such as 'eintreten' (to enter), 'ausüben' (to practice), or 'tätig sein' (to work/be active). The preposition is then usually 'in' or 'im'.
Nach dem Studium möchte er so schnell wie möglich ins Lehramt eintreten.
After his studies, he wants to enter the teaching profession as quickly as possible.
preparatory traineeship
'Lehramt' and 'Referendariat' both belong to the path into the teaching profession. 'Lehramt' means the profession of being a teacher or the course of study that prepares you for this profession. 'Referendariat' is the practical training period after university studies, during which you already work at a school and continue to be assessed. Learners may confuse the words because both come up in conversations about teacher training. Remember: first you study teaching, and after that usually comes the 'Referendariat' (preparatory traineeship).

Nach dem Abitur möchte Lena Lehramt studieren, weil sie gern mit Jugendlichen arbeitet.
After finishing school, Lena wants to study teaching because she enjoys working with teenagers.
Im Referendariat unterrichtet Paul zwei Klassen und besucht zusätzlich Seminare.
During his preparatory traineeship, Paul teaches two classes and also attends seminars.