And then
Ways to carry on to the next thing: -te inside a sentence; de, jaa and ato at the start of one.
-te
and (then); so
て
The te-form chains clauses: one action then the next, or a cause and its result ('tsukarete, neta'). Adjectives join with -kute, nouns and na-adjectives with de.
uchi kaette, neta.
I went home and slept.
うち帰って、寝た。
uchi ni kaette, nemashita.
I went home and slept.
うちに帰って、寝ました。
Tense and politeness go only on the last verb, in polite speech too: 'kaette, nemashita', not 'kaerimashite' (that sounds formal). A -te reason can't lead into a request or invitation ('isogashikute, tetsudatte' is wrong); use kara there.
de
polite soredeso; and then
で それで
Starts a sentence that moves a story to the next step, or prompts the speaker to go on ('de?' = 'and?'). Clipped casual form of sorede.
de, dou natta?
So, what happened?
で、どうなった？
sorede, dou narimashita ka?
So, what happened?
それで、どうなりましたか？
Comes first, after a pause; that's how to hear it apart from the particle de. Bare 'de?' can sound impatient; 'de, de?' or 'sorede?' with rising pitch sounds keen.
jaa
well then; so
じゃあ
Draws a conclusion from what was just said and moves to the next step: deciding, ordering, wrapping up, saying goodbye. Short form ja.
jaa, kore ni suru.
Okay then, I'll go with this.
じゃあ、これにする。
jaa, kore ni shimasu.
Okay then, I'll go with this one.
じゃあ、これにします。
To a clerk or coworker, jaa + masu is normal ('jaa, kore ni shimasu'); dewa is the formal version for meetings, announcements and shop staff.
ato
oh, and; also
あと
Starts a sentence that adds one more item to a list, order or request, often as an afterthought.
ato, kore mo onegai.
Oh, and this too, please.
あと、これもお願い。
ato, kore mo onegaishimasu.
Oh, and this one too, please.
あと、これもお願いします。
Ato de = 'later'; ato + amount = 'more / left' ('ato gofun' = five more minutes); bare ato at the start, then a pause = 'oh, and'.