〜て (and then)
…and (then)…, joining actions
Verb て-form, Verb
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The て-form joins actions in the order they happen: 朝起きて、顔を洗います, "I get up in the morning and wash my face".
Only the last verb shows tense and politeness; the earlier ones are all て-forms. 家に帰って、ご飯を食べました is "I went home and ate", past because of ました at the end.
It's built like the た-form, with て/で for た/だ: 食べて, 買って, 飲んで, 書いて, 泳いで, 話して, して, 来て (きて), and 行って.
The て-form is the most useful form in the language. The next several points (てください, ている, てもいい, てから…) are all built on it.
Examples
朝起きて、顔を洗います。
I get up in the morning and wash my face.
家に帰って、晩ご飯を食べました。
I went home and had dinner.
デパートに行って、服を買いました。
I went to the department store and bought some clothes.
本を読んで、寝ました。
I read a book and went to sleep.
シャワーを浴びて、出かけます。
I'll have a shower and head out.