Translation:Shall we go to the movie theater next month?
It's a slight difference but should doesn't usually function this way. It is used when you ARE SUPPOSED to do something, in this case you are proposing something so "shall" is much more appropriate. Treat it like a more polite version of "will" (future tense). So instead of asking "will you go out with me" or something you'd rather say "shall we mmet again" or "shall we go to te movies". Should would imply that it's some oin od a responsibility loo
Feel free to correct me if I'm wrong but this is my understanding so far: ... ~ませんか = do you want to ~... In this case you're trying to figure out if the other person is interested. So, for example, this would be like asking someone "Do you want to go to the ball with me?" ... ~ましょうか = shall we ~... In this case, the person is already interested, and you're basically asking for confirmation. Taking the previous example, this would be like if you're already at the ball and you're asking, "Shall we dance?"
You are absolutely correct, breazu. 〜ましょうか is also a more forceful proposition where the implication is more or less "We both know you're not going to refuse." Whereas ~ませんか is a more careful and more conservative approach. Hence, as far as I'm concerned, the "Do you want to ... ?" translation should not apply here if only so people who encounter this problem come to this thread to read up on it.
"e" means "towards" while "ni" means "to".
Eki e ikimasu. I go towards the train station (but may change the direction while I'm going).
Eki ni ikimasu. I go to the train station (and it is my definite destination).
While talking about places e and ni are often interchangeable.