zuerst
first
Indicates the first action in a chronological sequence; first of all.
Zuerst putzen wir die Küche und danach sehen wir einen Film.
First we clean the kitchen, and after that we watch a film.

first / at first
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first
Indicates the first action in a chronological sequence; first of all.
Zuerst putzen wir die Küche und danach sehen wir einen Film.
First we clean the kitchen, and after that we watch a film.

The word 'zuerst' (first / at first) is made up of the preposition 'zu' and the numeral 'erst' (from 'erste', first). It literally means 'as the first thing' or 'at the beginning'.
Wer zuerst kommt, bekommt den besten Platz.
Whoever comes first gets the best seat.
When 'zuerst' (first / at first) stands at the beginning of the sentence (position 1), the verb follows directly after it in position 2. However, the adverb can also stand in the middle of the sentence after the verb and subject.
Zuerst trinken wir einen Kaffee.
First we drink a coffee.
To describe a chronological sequence of actions, 'zuerst' (first) is very often combined with adverbs like 'dann' (then) or 'danach' (after that). This pattern structures a story clearly.
Zuerst machen wir die Hausaufgaben, dann gehen wir spielen.
First we do the homework, then we go play.
In the meaning of 'anfangs' (initially), 'zuerst' (at first) is often used to express a contrast. A signal for this is a following conjunction like 'aber' (but) or 'doch' (after all / however), which introduces a change of mind or a change of plan.
Zuerst fand ich den Film langweilig, aber am Ende war er doch sehr spannend.
At first I found the film boring, but in the end it was actually very exciting.
once / times
'mal' in 'erst mal' and 'zuerst' can both express that something happens at the beginning. 'zuerst' clearly describes the first action in a sequence: first A, then B. 'erst mal' sounds more casual and can also mean: for the moment, we are only doing this step. So it does not always have to refer to an exact sequence with several steps. In everyday speech, people often say 'erst mal' when they do not want to decide everything yet. 'zuerst' fits better when the order is important.

Vor dem Training wärmt sich das Team zuerst zehn Minuten auf.
Before training, the team warms up for ten minutes first.
Heute trinken wir erst mal einen Kaffee und planen dann den Einkauf.
Today we'll have a coffee first and then plan the shopping.