Why is the dropped in "I go to school by bus"? Why isn't it "I go to the school by the bus" if both school and bus are countable? Does the rule that a countable noun must have an article have an exception here?
There are a number of cases where nouns are not preceded by any article (‘the zero article’). They include meals and places as institutions, times of the day, days, months and seasons and, as here, means of transport and communication. We speak of going by air, car, horse or rail and sending by mail, post or e-mail.
School without an article means the school as an educational system. It's sort of like an abstract noun, I think.
A/the school places more importance on the physical aspect of school. (Not necessarily the case for "the".)
He is in school.
= He is a student.
He is in the school.
= He is in a school building.
I'll try to answer only the part about by bus. Adjunct prepositional phrases beginning with by, when they have the function of indicating manner/means, seem to require mass nouns, or derived nouns which behave like mass nouns in not needing an article.
I can think of three major categories of nouns appearing in the manner/means by-prepositional phrase. The first is gerundive verb phrases. Gerundive verb phrases can have two variants, one with no article, and an unmarked object, and another with an article, and the object "demoted" to an of-prepositional phrase. Note that only the article-less version is grammatical.
Columbo solved the crime by tricking the suspect.
*Columbo solved the crime by the tricking of the suspect.
Dawn hurt her knee by tripping over a rock.
Barney ruined his chances for college by drinking heavily.
The second category is deverbal nouns formed with the sundry latinate nominalizing suffixes:
Mason got money by litigation.
The iron rusts by oxidation.
Johnson gained agreement for his plans by deceit.
The home's value was determined by appraisal.
The third category is underived mass nouns:
Thomas can tell by smell whether the cookies are done.
Thomas made cookies by hand.
Thomas can recite the Koran by heart.
Thomas obtained signatures by force.
I think phrases like by bus and by foot would fall into this latter category. Though these nouns might not seem like mass nouns when you reflect on their referents, they are acting as mass nouns grammatically, because this particular construction only works with article-less noun phrases. To see that by bus contains a mass noun, compare it with the following odd-sounding sentence.
??Thomas goes to work by MARTA.
(MARTA: = metropolitan atlanta rapid transit authority)
The noun MARTA is just too specific to fit into the "grinder", but more general-sounding nouns like bus, horse, foot, car, etc., can do it.
The first answer is ok but let me add something. When we talk about the primary purpose of the institute or building where we go for our job or work, we do not use the article a or the. If one visits the place or institute other than the primary purpose, and article is used.
For example: A doctor goes to clinic. However, if his friend comes to see him there, the sentence would be: The doctor's friend visited him in the clinic, or, My father goes to the school to see my teachers, etc.
Why the is dropped in I go to school by bus instead of I go to the school by the bus.wherein school and bus are countable?
Why, is because it is not necessary to use the word the, when talking in a general sense. It is the same reason why people don't normally use it when talking about other situations, like going to university, going to hospital (although Americans make an exception for going to hospitals, so it is worth also seeing my answer here Is there a reason the British omit the article when they "go to hospital"?) or going to work. Use of the word 'the', means that the sentence is in a particular sense. A reference to one, particular example. "I go to the school by the bus" will mean going to one, particular school and travelling on one, particular bus.
If what you mean is attending a school in the general sense, without a particular need to say which one; then saying "I go to school", will be sufficient. If you also mean that you travel there on a bus or buses, but do not have a particular need to say which bus or buses; then saying "by bus", will be sufficient.
Saying "I go to the school" will mean going to one, particular school. Saying "I go to the school by the bus" will mean going on one, particular bus.
Saying "I go to the school by the bus" would only make sense if you have already said which, particular school and bus, you are talking about.
Basically, the word the is not used when talking in the general sense, because it gives the wrong impression, which is that you are talking in the particular sense. Using it would therefore be unnecessary and make your exact meaning, unclear.
It is not a case of the word 'the' being dropped. It is a case of it being unnecessary and giving a different meaning from what you intended.