anstoßen
to toast
To lightly clink glasses together in order to drink to someone or something; to make a toast.
Lasst uns auf das Brautpaar anstoßen!
Let's toast the bride and groom!

to toast / initiate / bump
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to toast
To lightly clink glasses together in order to drink to someone or something; to make a toast.
Lasst uns auf das Brautpaar anstoßen!
Let's toast the bride and groom!

'Anstoßen' (to toast/initiate/bump) is a separable verb with the prefix 'an-'. In a main clause, the prefix goes at the end. In subordinate clauses or in the infinitive with 'zu', the verb stays together.
Er stößt eine wichtige Diskussion an.
He initiates an important discussion.
When drinking to something or someone with glasses, you use the preposition 'auf' (to) followed by the accusative. This structure is firmly connected with this meaning.
Wir wollen auf das neue Jahr anstoßen.
We want to toast the new year.
In the meaning 'initiieren' (to initiate), 'anstoßen' is almost always used with abstract nouns. Typical examples are: 'eine Debatte anstoßen' (to initiate a debate), 'eine Reform anstoßen' (to initiate a reform), 'eine Veränderung anstoßen' (to initiate a change), or 'ein Projekt anstoßen' (to initiate a project).
Mit ihrer Rede wollte die Politikerin eine neue Klimadebatte anstoßen.
With her speech, the politician wanted to initiate a new climate debate.
When you hurt yourself, the verb is reflexive: 'sich anstoßen' (to bump oneself). The object you bump into is in the dative.
Ich habe mich im Dunkeln an der Türkante angestoßen.
I bumped myself on the edge of the door in the dark.
In the meaning 'angrenzen' (to border on), 'anstoßen' is always used with the preposition 'an' and the accusative. This meaning is rather formal or written.
Der Park stößt direkt an unser Firmengelände an.
The park borders directly on our company premises.
to drink
'auf etwas trinken' and 'anstoßen' both belong to a celebration or a special moment. With 'auf etwas trinken' (to drink to something), the focus is on celebrating someone or something with a drink. 'anstoßen' especially emphasizes the action with the glasses: you lightly clink them together. Often, both happen at the same time. That is why learners may confuse the words. If you are naming the wish or the occasion, 'auf etwas trinken' works well. If you mean the gesture with the glasses, 'anstoßen' is more precise.

Frau Demir hat mit ihrem Freund angestoßen, als die Reise gebucht war.
Ms. Demir toasted with her friend when the trip had been booked.
Lea trinkt mit ihren Freunden auf den neuen Job.
Lea drinks to the new job with her friends.