
besteigen
to climb
To climb up or ascend something, especially a mountain, a tower, or another elevated place.
Die Alpinisten wollen morgen den Gipfel des Mount Everest besteigen.
The mountaineers want to climb the summit of Mount Everest tomorrow.
to climb / board
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to climb
To climb up or ascend something, especially a mountain, a tower, or another elevated place.
Die Alpinisten wollen morgen den Gipfel des Mount Everest besteigen.
The mountaineers want to climb the summit of Mount Everest tomorrow.

to board
To get into a means of transport, such as a train, bus, ship, or airplane, or onto a riding animal, such as a horse, in order to travel with it.
Die Passagiere besteigen gerade das Flugzeug nach New York.
The passengers are currently boarding the plane to New York.

to ascend the throne
To take up rule as a monarch.
Nach dem Tod ihres Vaters bestieg die Prinzessin den Thron.
After her father's death, the princess ascended the throne.

to mount
(technical, biological) For an animal to mate with another animal for reproduction.
In der Dokumentation wurde gezeigt, wie der Hengst die Stute auf der Weide besteigt.
The documentary showed how the stallion mounted the mare in the pasture.
'Besteigen' (to climb/board) is an inseparable verb made up of the prefix 'be-' and the verb 'steigen' (to climb/rise). The prefix 'be-' makes the verb transitive, so it requires a direct object in the accusative.
Er hat schon viele hohe Berge bestiegen.
He has already climbed many high mountains.
After 'besteigen' (to climb/board), there is always an accusative object without a preposition. Say 'den Berg besteigen' (to climb the mountain), not 'auf den Berg besteigen'.
Wir wollen am Wochenende den höchsten Turm der Stadt besteigen.
We want to climb the tallest tower in the city this weekend.
In this meaning, 'besteigen' (to climb) is typically combined with high places. Common objects are: einen Berg (a mountain), einen Gipfel (a summit), einen Felsen (a rock), or einen Turm (a tower).
Nur erfahrene Kletterer sollten diesen Felsen besteigen.
Only experienced climbers should climb this rock.
When the object is a means of transport or a riding animal, 'besteigen' (to board/mount) means something like 'einsteigen' (to get in/board) or 'aufsteigen' (to mount). It is a somewhat more formal variant.
Die Reiterin bestieg ihr Pferd und ritt davon.
The rider mounted her horse and rode away.
The phrase 'den Thron besteigen' (to ascend the throne) is a fixed expression. It means that a person becomes king or queen.
Prinz Charles hat nach dem Tod der Queen den Thron bestiegen.
Prince Charles ascended the throne after the Queen's death.
to rise / climb
„Steigen“ and „besteigen“ can both describe going up a mountain. With „steigen“, the destination is usually given in a directional phrase such as „auf den Berg“. „Besteigen“, by contrast, is transitive: the mountain appears directly as an accusative object.

Die Wanderer besteigen am Morgen den Berg.
The hikers climb the mountain in the morning.
Die Wanderer steigen am Morgen auf den Berg.
The hikers climb the mountain in the morning.