wandern
to hike
To cover a longer distance on foot in nature, usually as a form of recreation or sport.
Am Wochenende gehen wir gerne in den Bergen wandern.
On weekends, we like to go hiking in the mountains.

to hike / migrate / wander
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to hike
To cover a longer distance on foot in nature, usually as a form of recreation or sport.
Am Wochenende gehen wir gerne in den Bergen wandern.
On weekends, we like to go hiking in the mountains.

Since 'wandern' (to hike / migrate / wander) describes movement from one place to another, the perfect tense is always formed with the auxiliary verb 'sein' (to be). Using 'haben' (to have) here is a common mistake.
Wir sind letztes Wochenende durch die Alpen gewandert.
We hiked through the Alps last weekend.
The verb 'wandern' (to hike / migrate / wander) belongs to a word family connected with walking in nature. Related words are 'die Wanderung' (the hike) and 'der Wanderer' (the hiker).
Für die geplante Wanderung brauchen wir gutes Schuhwerk.
For the planned hike, we need good footwear.
In the meaning 'spazieren gehen' (to go for a walk), 'wandern' (to hike) is often used in the combination 'wandern gehen' (to go hiking). Typical location phrases are 'in den Bergen' (in the mountains), 'im Wald' (in the forest), or 'am See entlang' (along the lake).
Am Sonntag gehen wir im Schwarzwald wandern.
On Sunday, we are going hiking in the Black Forest.
When the subject of the sentence is something abstract or an object that cannot move by itself, the figurative meaning is usually intended. Typical subjects are 'der Blick' (the gaze), 'die Gedanken' (the thoughts), or 'ein Gegenstand' (an object).
Seine Gedanken wanderten zurück in seine Kindheit.
His thoughts wandered back to his childhood.
The meaning 'migrieren' (to migrate) is usually recognized from context. It is used almost exclusively for animal species (e.g., birds, fish) or nomadic peoples that change their habitat.
Die Störche wandern im Herbst in den Süden.
The storks migrate south in autumn.
to run
'laufen' and 'wandern' both mean that you move on foot. 'laufen' can mean walking fast or running; colloquially, it can also simply mean 'to go on foot'. 'wandern' is used for longer routes in nature, usually for recreation or as a sport. Learners may confuse the words because when you hike, you also walk. But 'wandern' says more about the situation: it is longer, often outdoors in the mountains, forest, or countryside. 'laufen' is more general and can also fit everyday situations or sports.

Am Sonntag wandern wir mit Freunden durch den Wald.
On Sunday, we are hiking through the forest with friends.
Hannes läuft zum Bus, weil er spät dran ist.
Hannes runs to the bus because he is late.