Grammar JLPT Level N5 - Lesson 10

In Japanese, て is a very important conjunction particle that can be used with many different structures. When it is partnered with a verb, it has special conjugation rules, depending on whether it is being added to a る-Verb or a う-Verb. In the case of う-Verbs, it also depends on what the preceding kana is.
The most common translation in any situation is just 'and' or 'then', due to (B) being highlighted as something that happens/happened after (A).

• 山(やま)を登(のぼ)って休(やす)む。
  To climb a mountain, then rest. (To climb a mountain, and rest)
• 食(た)べて寝(ね)る。
  To eat, then sleep. (To eat, and sleep)

In these examples, we can see that て basically means '(A) て (B)' = '(A) happened, then (B)'. However, this is only when it is linked to another verb.
Sometimes the て form appears as で. This is a change that happened throughout the course of history, in order to make sentences flow more smoothly. Despite this, there is no difference in meaning between て and で as a vocal change.

• 本(ほん)を読(よ)んで返(かえ)す。
  To read a book, then return it. (て appearing as で)
• 遊(あそ)んで帰(かえ)る。
  To play, then go home. (て appearing as で)

The easiest way to identify whether to use て or で is by looking at the preceding kana. If the plain (dictionary) form of the verb finishes in ぐ, ぬ, ぶ, or む, then で will be used. る-Verbs never use で.