Shorts (nice ones) are usually okay on the first night. People are still tired form travel and not everyone has their luggage so it is generally very casual.
Not knowing how old your kids are .... for my daughter, every night is formal night. For my son, he wears a golf shirt and long walking shorts on the first night. For causal nights, he will wear a button down shirt with dress pants from the children's place with dress shoes. For formal nights he wears the same dress pants with a shirt/tie combo. [my son is 4]