Как правильно написать I would like to see/seeing the Grand Canyon one day.

Ответ: I would like to see the Grand Canyon one day. Пояснение: После would like (хотел бы) мы используем неопределённую форму  с частицей to. А вот после like можно употреблять как инфинитив с to, так и с герундием (ing-form) без to. Ср.: I would like to go to that city. Elisabeth likes visiting old towns. Alex likes to walk alone.