Translation:I eat breakfast in the cafeteria.
I believe that either works. The only time I can see any real/meaningful difference would be if I went somewhere that had outdoor seating, I would say "I eat breakfast at the cafeteria", because I'm not technically inside the cafeteria, I'm just in the close vicinity.
Other than that, I usually prefer "in" for generic locations (in the cafeteria, in the mall, etc.), but prefer "at" for specific locations (at McDonalds, at Red Robin). No reason why, its just what came out of my mouth when I rambled off a few sentences to see what I naturally do.
As far as trying to improve the little nuances of English, I think that non-natives can use either without raising an eyebrow from anyone. =D
According to my Japanese dictionary, 食堂 can either mean a room where you eat, or a place (store) where you can eat. Basically this can mean a cafeteria in an institution, but I think it's seldom used to refer to a Western restaurant. If I hear this word I always think of a small house offering some simple traditional Japanese food. I hope someone can confirm this.
I eat breakfast in the cafeteria
we eat breakfast in the cafeteria
they eat breakfast in the cafeteria
he eats breakfast in the cafeteria
she eats breakfast in the cafeteria
These are all valid as the subject isn't stated, it's assumed we know it by context. So without context, all of these are correct. Unlike European languages or English, there is not first person singular, plural, second person, third person.... conjugation. In Japanese the verb conjugation for all is the same.
Both particles are pretty complicated and have a few different usages, so I'm going to focus on how to use に or で for locations. My answer would get even longer otherwise f(^_^;
I like to call に the "target" particle because in general, it tells you the direction or destination of a (movement) verb. In most cases, this translates to "to" but not all the time. Some examples:
学校に行きます。 (がっこにいきます) = "I go to school." 「学校」 (school) is the end point or target of the movement verb, 「行きます」 (to go)
このエレベーターは上に参ります。 (このエレベーターはうえにまいります) = "This elevator is going up." 「上」(up) is the direction of the movement verb, 「参ります」 (humble form of "to go")
銀行に送ります。 (ぎんこうにおくります) = "I'll send it to the bank." 「銀行」(bank) is the target of the non-movement verb, 「送ります」 ("to send")
On the other hand, I like to call で the "context" particle, in the sense that it is used to give you extra context for the verb. Generally, when attached to a location, で tells you where a non-movement verb is taking place. This usually corresponds to "at" or "in" in English, but not always. Some examples:
家で勉強します。 (いえでべんきょうします) = "I study at home." 「家」 (home) is where the verb, 「勉強します」 (to study), takes place.
自分の部屋で踊ってください。 (じぶんのへやでおどってください) = "Please dance in your own room." 「自分の部屋」 ("one's own room") is where the verb, 「踊る」 ("to dance") is requested to take place.
(!!) 成田空港で行きます。 (なりたくうこうでいきます) = "I'll go through Narita Airport." Here で is acting with a movement verb, so it behaves differently. Here it tells you "by what means" the movement verb is being done (more commonly used for describing the mode of transportation, but it can also be used this way). Notice that it still gives extra context, like it did for non-movement verbs.
Yes, verbs in their ます forms are simple present tense, not imperative (command).
Be aware though, that we assume we are referring to ourselves because we do not get any other information to suggest otherwise. This sentence can just as easily be translated to "He eats at the cafeteria" or "They eat at the cafeteria", depending on the context you use it in.
A "shyokudo" is a diner, usually a 2-story house (rural or suburbs, I never got to downtown...) where the family lives upstairs, and the ground floor is a small "restaurant". In 65 years, I've only heard "canteen" used in a military context (except "cantina" in Mexico and Tax-ifornia), and "dining room" at home or a hotel, or "dining hall" in a college dorm or other institution, "cafeteria" is a self-serve food line with trays and self-bussing your own table... A diner, or maybe loosely "cafe" is a better meaning of "shyokudo". Or maybe "greasy spoon", or "lunch joint" !?
This really needs cleaned up. A few seconds ago I was marked wrong for NOT translating リストランでばんごはんを食べます as "I eat MY dinner in a restaurant"; now I'm marked wrong for putting IN the "my" and translating this sentence, identical in structure, as "I eat my breakfast in a cafeteria"!
No, で indicates the location at which an action occurs. The "action" in this sentence is あさごはんを食べます or "to eat breakfast".
In your suggestion, the main "action" is "to get breakfast". The "to eat at the cafeteria" is actually an additional clause which is separate from the main "action"; it becomes clearer if you replace it with "for the purpose of eating at the cafeteria", which retains largely the same meaning.
Translating your suggestion into Japanese would look like this instead:
- "for the purpose of eating at the cafeteria, I get breakfast."
食べます becomes 食べる because it's not the main verb of the sentence.