Work and study

学 (xué) means to learn. Look at what it builds:

学生 (xué shēng) is a student, a learning-born-one. 学校 (xué xiào) is a school. 学习 (xué xí) is to study. 大学 (dà xué) is a university, a big school.

Four high-frequency words from one character, and you can now guess at others. 中学 is secondary school, middle-school. 小学 is primary. 同学 is a classmate, a same-learner.

This is the argument for learning characters rather than treating Mandarin words as opaque sounds. The sounds are arbitrary; the characters are not.

老 is not always old

老师 (lǎo shī) is a teacher. 老板 (lǎo bǎn) is a boss. Both start with 老 (lǎo), which means old, and in neither case is age involved.

老 attached to a role is a marker of respect and familiarity rather than a description. A twenty-two year old teacher is still 老师.

It works the same way on surnames: calling a colleague 老王 is warm and slightly informal, roughly the register of using someone's first name in a British office.

工作 is both

工作 (gōng zuò) is work as a noun and to work as a verb, with no change of form. Mandarin does this constantly, because it has no inflection to change: the same word simply occupies whichever slot the sentence gives it.

That is a large part of why Mandarin grammar is genuinely easier than European grammar, and it is worth saying plainly, because the writing system gets all the attention and gives the impression the whole language is hard.

教 has two readings

教 as a verb, to teach, is jiāo. In the compound 教育 (jiào yù), education, it is jiào.

Characters with more than one reading are called polyphones and there are a few hundred that matter. 教 is one of the first you will meet. The reading follows the grammar: the verb takes the first tone, the noun-forming compound takes the fourth.