영화 볼래요? ( Will you watch the/a movie?) 영화 볼까요? (Shall we watch the/a movie?) 정은 씨, 같이 식당에 갈래요? (Jung-eun, will you go dine with me?) 정은 씨, 같이 식당에 갈까요? (Jung-eun, shall we go dine together?) 정은 씨, 이번 연휴에 같이 여행갈래요? (Jung-eun, will you go travelling with me this coming holiday?) 정은 씨, 이번 연휴에 같이 여행갈까요? (Jung-eun, shall we go travelling together this coming holiday?) Hope you find this helpful. Note how the subject of each question changes by having either -래요 or -까요 at the end. * Please also bear in mind that --ㄹ까요 question structure also has other functionalities other than the one mentioned above, and is not thereby limited in its usage as such.

As you probably know, the actual meaning of these two verb endings is basically the same. However, they have a slightly different nuance. Usually, ㄹ래요 is asking the person a question, but ㄹ까요 is suggesting that the speaker and the other person do something together. For example: 물 마실래요? - Do you want to drink water? 물 마실까요? - Shall we drink water (together)? In two of your examples, there is '같이' which means the meanings of both are the same.

-ㄹ래요 = Do you want to ... ? -ㄹ까요 = Shall we ...? -ㄹ래요 asks specifically for what the listener wants to do, while -ㄹ까요 is just a general suggestion.