This summer, for the first time in a long time, I plan to go on a trip with my mother, my big brother's family, and my little sister's family.
ご質問ありがとうございます。
"This summer, for the first time in a long time"＝「この夏、久しぶりに」
"I plan to go on a trip with my mother, my big brother's family, and my little sister's family."＝「私は母、兄の家族、そして妹の家族と旅行に行く予定だ。」
ご参考に。
アメリカ合衆国 
