einer Sache nachgehen
to pursue
To engage in a particular activity, hobby, or profession.
In seiner Freizeit geht er dem Angeln nach.
In his free time, he pursues fishing.

to pursue / follow up
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to pursue
To engage in a particular activity, hobby, or profession.
In seiner Freizeit geht er dem Angeln nach.
In his free time, he pursues fishing.

'Nachgehen' (to pursue / follow up) is a separable verb. The prefix 'nach-' comes at the end in a main clause. In subordinate clauses and infinitive constructions, the verb stays together.
Er geht seiner Leidenschaft für die Fotografie nach.
He pursues his passion for photography.
In most meanings, 'nachgehen' (to pursue / follow up) requires the dative. You go after 'einer Sache' (a thing) or 'jemandem' (someone), not 'eine Sache' or 'jemanden'.
Der Detektiv geht einer neuen Spur nach.
The detective is following up a new lead.
When it is about activities, 'nachgehen' (to pursue) is often combined with 'einem Hobby' (a hobby), 'einer Tätigkeit' (an activity), 'einer Leidenschaft' (a passion), or 'seinem Beruf' (one's profession).
Nach der Arbeit geht sie ihrer Leidenschaft, dem Malen, nach.
After work, she pursues her passion, painting.
In the context of investigations, 'nachgehen' (to follow up) is often used with 'einem Hinweis' (a lead), 'einer Spur' (a trail/lead), 'einem Verdacht' (a suspicion), or 'einer Frage' (a question).
Die Journalistin will der Sache weiter nachgehen.
The journalist wants to follow up the matter further.
You can recognize this meaning by the fact that the subject of the sentence is a clock, watch, or another timekeeping device. Here the verb is used without an object.
Die alte Bahnhofsuhr geht schon seit Jahren nach.
The old station clock has been running slow for years.
In the meaning 'folgen' (to follow), 'nachgehen' (to follow someone) always appears with a person (or an animal) in the dative. It describes concrete, physical movement.
Der Hund ging seinem Besitzer überallhin nach.
The dog followed its owner everywhere.
to follow
'folgen' and 'nachgehen' can both mean that someone goes after a person or follows a trail. 'folgen' is the normal word when you move behind someone. 'jemandem nachgehen' can also mean this, but it often sounds more purposeful or observant. In addition, 'einer Sache nachgehen' often means to investigate something, for example a lead or a question. Learners may confuse the words because both express a direction behind someone or something. For simply going after someone, 'folgen' is usually more natural. For investigating or following up on something, 'nachgehen' fits better.

Die Polizei geht einem neuen Hinweis nach.
The police are following up on a new lead.
Du folgst einfach dem Weg bis zur Brücke.
You simply follow the path to the bridge.