저 is the polite form of 나. Both words mean "I". If 저 or 나 meet the subject particle (-가), they become 제가 or 내가. (we're kind of adding one stroke of 이 there) When to use: In short, if people ask you "Where do you want to go?" you could answer with: 저는 한국에 가고 싶어요. 나는 한국에 가고 싶어요 But if people ask "Who want to go to Korea?" you answer with: 제가 한국에 가고 싶어요. 내가 한국에 가고 싶어요. Use the polite forms (저 or 제가) to people who you don't know well or to the ones who you respect.

correct! It's like the sentences you wrote 제가 (polite form) 저는 as subject/저를 as object (polite form) 나는 (casual form) 내가 (casual form)