
danach
afterward
Indicates a point in time that follows a previously mentioned event; afterward, later.
Wir essen erst zu Abend und gehen danach ins Kino.
We'll have dinner first and go to the movies afterward.
afterward / after that
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afterward
Indicates a point in time that follows a previously mentioned event; afterward, later.
Wir essen erst zu Abend und gehen danach ins Kino.
We'll have dinner first and go to the movies afterward.

after it / for it
As a pronominal adverb, it refers to a thing or situation connected with the preposition 'nach' (for example 'fragen nach' (to ask about/for), 'riechen nach' (to smell of/like), 'sich sehnen nach' (to long for)).
Das riecht gut. Wonach duftet es hier eigentlich?
That smells good. What does it actually smell like in here?

accordingly
Expresses a logical result or consequence; accordingly, consequently.
Er hat die Regeln gebrochen und wird auch danach beurteilt.
He broke the rules and will also be judged accordingly.
'Danach' (afterward/after that) is a pronominal adverb made up of 'da' and the preposition 'nach'. Here, 'da' functions like a pronoun and refers to something that was mentioned earlier.
Er fragte nach dem Weg. Ich konnte ihm danach aber nicht helfen.
He asked for directions. But I couldn't help him with that.
When 'danach' (afterward/after that) stands at the beginning of a main clause (position 1), the conjugated verb follows directly in position 2. This word order is called inversion.
Zuerst trinken wir Kaffee. Danach machen wir einen Spaziergang.
First we'll drink coffee. Afterward we'll take a walk.
As a pronominal adverb, 'danach' replaces the combination 'nach + Sache' (after/for + thing). It is often used with verbs that have the fixed preposition 'nach', such as 'fragen nach' (to ask about/for), 'suchen nach' (to search/look for), 'sich sehnen nach' (to long for), or 'riechen nach' (to smell of/like).
Sie sehnt sich nach Urlaub. Ich sehne mich auch danach.
She longs for a vacation. I long for it too.
then
„Dann“ and „danach“ can both express a sequence in time. „Dann“ usually simply continues with the next step in a story, while „danach“ explicitly refers back to an event mentioned earlier. Use „dann“ for the sequence of events and „danach“ for a clear reference to the event.

Wir kochen gemeinsam, und danach essen wir im Garten.
We’ll cook together, and afterward we’ll eat in the garden.
Zuerst kaufen wir ein, dann kochen wir gemeinsam.
First we’ll go shopping, then we’ll cook together.