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.

'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 chronological sequence. In many everyday sentences, they mean almost the same thing: first something happens, and later something else happens. 'danach' refers a little more clearly to a previously mentioned event. 'dann' is more general and also works well when listing steps in an action. In the meaning of 'afterward', the two can often be exchanged.

Herr Weber hat im Garten gearbeitet, danach hat er sich einen Tee gemacht.
Mr. Weber worked in the garden; afterward, he made himself some tea.
Zuerst räume ich die Küche auf, dann gehe ich einkaufen.
First I'll tidy up the kitchen, then I'll go shopping.