fischen
to fish
To catch fish or other aquatic animals from the water, usually with a fishing rod, a net, or other tools.
Am Wochenende gehen wir oft am See fischen.
On weekends, we often go fishing at the lake.

to fish
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to fish
To catch fish or other aquatic animals from the water, usually with a fishing rod, a net, or other tools.
Am Wochenende gehen wir oft am See fischen.
On weekends, we often go fishing at the lake.

The verb 'fischen' (to fish) is derived directly from the noun 'der Fisch' (fish). It describes the activity one does in order to catch fish.
Mein Opa hat mir beigebracht, wie man Fische fängt und richtig fischt.
My grandpa taught me how to catch fish and fish properly.
The preposition after 'fischen' (to fish) changes the meaning strongly. Without a preposition or with a place ('am See' — at the lake), it means fishing. With 'aus' (out of), it means to pull something out, and with 'nach' (for), it means to look for something indirectly.
Er fischt eine Münze aus seiner Hosentasche.
He fishes a coin out of his pants pocket.
In the meaning 'etwas herausholen' (to pull something out), 'fischen' (to fish out) is almost always used with the preposition 'aus' (out of) and the dative. This construction describes where something is taken from.
Sie fischte die Nudeln mit einer Gabel aus dem heißen Wasser.
She fished the noodles out of the hot water with a fork.
The figurative meaning 'nach etwas fragen' (to ask for something) works only with the preposition 'nach' (for). Typical nouns in this expression are compliments, information, or recognition.
Er stellt unauffällige Fragen und fischt so nach Informationen über das Projekt.
He asks discreet questions and thus fishes for information about the project.
to catch
'fangen' and 'fischen' can both mean that you get something and take it out of its surroundings. 'fangen' is more general: you can catch a ball, an animal, or a person. 'fischen' is more specific when it is about fish or other animals in the water. You also say 'etwas aus etwas fischen' (to fish something out of something) when you pull an object out of water or out of a cluttered place. If you only want to say that you hold onto or get hold of something, 'fangen' fits. If water, fish, or pulling something out is the focus, 'fischen' sounds more natural.

Lisa fischt ihren Schlüssel aus der tiefen Tasche.
Lisa fishes her key out of the deep pocket.
Beim Training fängt Max den Ball kurz vor der Linie.
During practice, Max catches the ball just before the line.