"I take the subway."
The short answer is, the particle depends on what verb you are using (and what kind of verb it is).
The longer answer is that the information above is not correct, which might be what's confusing you. If I say "I go to school by car", it's 車で学校に行きます (kuruma de gakkou ni ikimasu). "I go to school by bike" is 自転車で学校に行きます (jitensha de gakkou ni ikimasu). See some examples at weblio.
This sentence isn't saying that you are going somewhere by subway, it is saying that you take the subway. In English "the subway" would be the direct object of the verb, meaning the verb "take" is a transitive verb, a verb capable of taking a direct object. I take what? I take the subway.
Unfortunately, the Japanese verb 乗ります (norimasu) is an intransitive verb, meaning that it cannot take a direct object. It's more naturally translated as "take" in English, but it more literally means "ride". To say what you are riding in/on, you need the particle に. 地下鉄に乗ります (chikatetsu ni norimasu).
乗ります (norimasu) is often translated as "ride", e.g. 自転車に乗ります (jitensha ni normasu) is "ride a bike". When it comes to transportation like a subway, saying "ride the subway" is correct English, but it's more common to say "take the subway". The words are synonyms. "Use" has a similar meaning, but has an exact Japanese equivalent.
Chikatetsu ni norimasu.
I take the subway. / I ride the subway.
Chikatetsu o tsukaimasu.
I use the subway.
I'll try to keep fancy grammar terms to a minimum here.
は、を、and に are all something called particles. A particle is a "word" (or word part) so basic that it carries no meaning on its own, and so always appears next to another word to give it meaning. An example of this is in English is apostrophe-S for possession--as in "John's book." Here, the apostrophe-S is meaningless unless it is paired with it's companion, John. When joined with John, it becomes and adjective meaning "belong to ___."
Japanese has a lot of particles, which is one part of what makes it so hard for European language natives to learn.
Let's break down は、を、に：
は is one of the most common particles. It's called the topic particle. It can be read as "as for (the thing before me)." For example: 電車は大きいです could be understood as "as for trains, they are big."
を is known as the object particle. It marks the thing before it as the receiver of the action in a sentence. やさいを好きです could be read as "vegetables are liked (by me)" because the vegetables are the things receiving the liking.
に is a particle that shows the intended audience of communication or means by which an action is carried out. あなたを電話にはきます means "to you, by (using) the telephone, i talk." The telephone uses に because it is the thing you use to talk.
I hope that clears up their differences a bit! :)
を - pronounced 'o' when used as a particle - indicates that an object is the target of a verb. は - 'wa' when used as a particle - indicates that the preceding clause is the topic that we are speaking about. に - indicates that you do something 'by way of' the preceding thing, such as a place or a method of transport.
"に - indicates that you do something 'by way of' the preceding thing, such as a place or a method of transport."
Not quite. で is used (among its other uses) to indicate the method of transport. It's simply that 乗る（のる） is an intransitive verb in Japanese, so に needs to be used to indicate the thing you're getting in/on.