Breakdown of Tu devrais d'abord saluer le chef.
tu
you (informal singular)
le chef
the boss / the manager
devoir
to have to / must
le
him, it (masculine direct object pronoun)
Questions & Answers about Tu devrais d'abord saluer le chef.
What is the difference between tu dois and tu devrais?
Tu dois is the present tense and means 'you must' or 'you have to' — it is a strong obligation. Tu devrais is the conditional form, meaning 'you should'. It is much softer and is the standard way to give advice in French, just like in this sentence.
Why does d'abord come before saluer? In English we say 'greet the boss first'.
In French, adverbs like d'abord (first / first of all) typically go directly after the conjugated verb. Here, that conjugated verb is devrais. So the French structure is literally 'You should first greet...'.
Why do we use d'abord for 'first' instead of premier?
Why isn't the verb saluer conjugated?
When you have two verbs together where the second depends on the first, only the first verb is conjugated. Because we already conjugated devoir into devrais, the second verb saluer (to greet) stays in its dictionary form (the infinitive).