> 学校＿行きます。  
>  がっこう＿いきます。

I assume that it does, because で or を doesn't really makes sense. Since で
indicates that I'm going in school, and 学校 isn't really a direct object. Or is
it?
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grammar particles particle-に particle-を particle-で
Should I use the particle に for 学校＿行きます？
学校へ行きます is more common.

When using "へ" you are referring to a place/direction, while "に" you could use
for other verbs.

  * その目的地へと（行く）。
  * その目的地に何かありましたっけ？

People often use に because they are assuming they are entering the building
"School".

Go to cinema:

  * 映画館へ行く。Correct.
  * 映画館に行く。Correct.(Edited)

Entering the car:

  * 車へ入る。Not common.
  * 車に入る。Correct.
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You can use either に or へ here. As mentioned in @EddieKal's comment, both of
the following are correct:

> 学校 **に** 行きます。  
>  学校 **へ** 行きます。

  * this に is "a particle which indicates a place toward which [someone] or [something] moves" [1]
  * this へ is "a particle that indicates the direction toward which some directional movement or action proceeds" [2]

In general, you can use the above two interchangeably [3]:

> In actuality, native speakers use へ and に (of point of contact) almost
> interchangeably except in the case of
>
> ニューヨーク{__}の便はもうありません。  
>  There isn't any flight to / as far as New York any more.
>
> 〇 ニューヨーク **へ** の便はもうありません。  
>  〇 ニューヨーク **まで** の便はもうありません。  
>  ✖ ニューヨーク **に** の便はもうありません。(に cannot be followed by の)

An example:

> 東京 **に** 十二時に着いた。  
>  東京 **へ** 十二時に着いた。  
>  I arrived in Tokyo at 12:00 o'clock.

* * *

[1] A Dictionary of Basic Japanese Grammar p. 302 (ni7 に)  
[2] A Dictionary of Basic Japanese Grammar p. 116 (e へ)  
[3] A Dictionary of Basic Japanese Grammar p. 117 (e へ Related Expressions
section I)
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