Two things at once, every time, the very moment, and staying in an unchanged state — ながら vs 間に, たびに vs ごとに.
〜ながら V-ます stem + ながら
V-ます stem + ながら. Means 'while doing', two actions by the same person at the same time. The main action comes second.
音楽を聞きながら、勉強します。 I study while listening to music.
〜まま V-た/V-nai + まま; N + のまま; い-adj + まま; な-adj + なまま
V-た / V-nai / N + の / い-adj / な-adj + な + まま. Means 'as is, in that unchanged state' while another action happens.
電気をつけたまま、寝てしまった。 I fell asleep with the light on.
〜と すぐ V-dict + と、すぐ(に) + clause
V-dict + と、すぐ(に). Means 'right after doing X, Y happens'. Often describes habits or a quick sequence of actions.
家に帰るとすぐ、シャワーを浴びます。 As soon as I get home, I take a shower.
〜とたんに V-た + とたん(に)
V-た + とたん(に). Means 'the moment / as soon as' something happened, something unexpected followed. Used for past events.
窓を開けたとたんに、鳥が入ってきた。 The moment I opened the window, a bird flew in.
〜たびに V-dict + たびに / N + のたびに
V-dict / N + の + たびに. Means 'every time' or 'whenever' something happens, something else also happens.
この歌を聞くたびに、学生時代を思い出す。 Every time I hear this song, I remember my school days.
〜ごとに N / V-dict + ごとに
Noun / V-dict + ごとに. Means "every / each", for regular intervals or each member of a group.
この薬は六時間ごとに飲んでください。 Please take this medicine every six hours.