My first au pair experience

In 2010 I moved to Germany near Mainz and I became an au pair. (Perfect!) 
It was one of the best periods in my life, with three energetic, smart and beautiful little girls.  (I'm not sure which superlative you'd like to use, but add one, e.g., "best", "most rewarding", etc., to "one of the ...") 
When I arrived in Germany the girls were 4,5 and 6 years old. (Or "when I came to Germany...")

The first three weeks were a hard challenge for me because I couldn't speak German, and so I had must to learn quickly.  (You can just say "a challenge" because a challenge is in itself difficult.  Note "I had to (main verb)" without "must".)
My work schedule was different every day different and sometimes I worked during the night.  (You might mean "during the evening"; "during the night" would mean from midnight to 5 am, for example.) 
One of the best things that I liked was going to the swimming pool once a week with one kid. (You can also say "One of my favorite activities was going..."  Also, "one of the children" is probably more common.)

Great work using the past tense especially!