I suppose that Euler's first language was German. Probably the Swiss German dialect, which they speak in Basel.
I say this because: a) he was born and went to school and to the university in the German-speaking part of Switzerland (Basel), b) His mother's (Brucker) and brother's (Johann Heinrich) names sound definitely German, and c) he had a German wife (Katharina Gsell whom he met in St. Petersburg), d) He was a Protestant and his father was a Protestant pastor, e) Euler is a German name. There is little doubt that he also learned
French in his childhood, (his thesis is written in French) as well as Latin, Greek and Hebrew at school. Later he learned Russian and English. There are indications that he preferred German in daily life.
Following the custom of the time, he wrote his mathematical papers in Latin, papers on other subjects in German and French.
Here is a reference that confirms my guess:
Leonhard Euler and the Bernoullis: Mathematicians from Basel
By M. B. W. Tent, p. 210.