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あとで (ato de)means “after doing”.  To express that something happens after something else, we should use the past (ta-form) of a verb followed by あとで (ato de).  

昼ごはんの後で友達とお茶を飲みました。

授業の後で映画に行きませんか？

薬を飲んだ後で、一時間寝ました。

私は宿題をした後でテレビを見ます。

子供が寝た後で、映画を見みました

たあとで also have the nuance of emphasising the fact that action B takes place after A (and not before). It’s good to remember that this grammar point uses the た form, which is the past tense in plain form.

ほんをよんだあとで、ねました　

てから is used when the second action (B) is going to happen straight after the first (A). In a lot of A てから B sentences, action B is only possible after completing action A. For that reason, it is useful when you want to express actions that take place in a specific order, such as in your daily routine.

ほんをよんでから、ねました

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