Great question and you're likely to get differing opinions. I think most will agree that it's gramatically correct to use them for each item but some people don't. You can see them skipped in newspapers so it's acceptable for the purporse of saving space. In this case, it doesn't sounds as bad. But if you sad "I had a banana, apple and grapefruit for breakfast" I would probably be left wondering "a banana, apple and grapefruit what".

I agree with Logan and Blip. You can omit the additional articles in American English. From the standpoint of formal writing, it might be seen as being better to include them though. Using Logan's sentence pattern, there is also this example: For breakfast, I had a bagel, muffin, slice of cheesecake, and fruit. which is the same as For breakfast, I had a bagel, a muffin, a slice of cheesecake, and fruit. Note that fruit means fruit in general here though. It could a banana and an apple, or whatever. It isn't specified and you shouldn't really put "a fruit" when you are speaking about fruit in general because it is a mass noun. :)

You don't have to, "You can go to the cinema, theater, and museum." is fine.