In the Japanese language, て serves as an essential conjunctive particle that connects various grammatical structures. When used with a verb, it follows a specific set of conjugation rules that change depending on the verb type, whether it's a る-Verb or a う-Verb. For う-Verbs specifically, the conjugation also hinges on the final kana of the verb's dictionary form.

Because (B) is presented as an action that takes place after (A), the most frequent translation for this structure is simply 'and' or 'then'.

• 家(いえ)に着(つ)いて手(て)を洗(あら)う。
  To arrive home, then wash your hands. (To arrive home, and wash your hands)
• 朝(あさ)起(お)きてシャワーを浴(あ)びる。
  To wake up, then take a shower. (To wake up, and take a shower)

As seen in these examples, て fundamentally means '(A) て (B)' = '(A) occurs, followed by (B)'. This sequential meaning, however, is primarily when it is used to link to another verb.

On occasion, the て form will manifest as で. This is a sound change that occurred over time to allow for a more natural pronunciation. Functionally, there is no distinction in meaning between て and its voiced counterpart, で.

• 荷物(にもつ)を運(はこ)んで置(お)く。
  To carry the luggage, then put it down. (て appearing as で)
• プール(ぷーる)で泳(およ)いで帰(かえ)る。
  To swim at the pool, then go home. (て appearing as で)

The simplest method to determine if て or で should be used is to check the final kana of the verb. If the dictionary form of the verb ends in ぐ, ぬ, ぶ, or む, you will use で. It is important to note that る-Verbs will never use で.