The te-form: changes by verb ending
The te-form supports requests, connections, and state expressions. Group its changes by verb ending, but do not treat the te-form itself as past tense.
Group 1 sound changes
Group 1 verbs ending in う, つ, or る take って: 買う → 買って, 待つ → 待って, 帰る → 帰って. Endings む, ぶ, and ぬ take んで: 読む → 読んで, 遊ぶ → 遊んで, 死ぬ → 死んで.
く changes to いて, as in 書く → 書いて; ぐ to いで, as in 泳ぐ → 泳いで; and す to して, as in 話す → 話して. The common exception 行く becomes 行って. Check the voiced で in 泳いで.
Group 2 and irregular verbs
For group 2, remove る and add て: 食べる → 食べて and 見る → 見て. 帰る is group 1, so applying this rule to produce 帰て is incorrect.
Learn する → して and 来る → 来て（きて） separately. When forming a new verb's te-form, first identify its group and then check its ending within that group.
Read through to the sentence ending
朝ご飯を食べて、学校へ行きます。 means 'I eat breakfast and go to school.' Here it connects successive actions. The te-form can also express a reason or accompanying state, so it does not always impose a chronological sequence.
読んでください requests that someone read; 読んでいます can describe reading in progress. What follows the te-form determines the sentence's function, so do not stop interpreting at the te-form itself.
Check: What are the te-forms of 行く and 泳ぐ?
Show the answer and why
They are 行って and 泳いで. 行く is an exception to the usual く pattern, while the ぐ in 泳ぐ changes to いで.