vorziehen
to prefer
To like or prefer something or someone more than another person or thing.
Ich ziehe Tee dem Kaffee vor.
I prefer tea to coffee.

to prefer / move up / draw
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to prefer
To like or prefer something or someone more than another person or thing.
Ich ziehe Tee dem Kaffee vor.
I prefer tea to coffee.

'Vorziehen' (to prefer / move up / draw) is a separable verb with the prefix 'vor-'. In a main clause, the prefix comes at the end. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Ich ziehe einen ruhigen Abend zu Hause vor.
I prefer a quiet evening at home.
When expressing a preference, 'vorziehen' (to prefer) has a fixed structure: You prefer [something in the accusative] to [something else in the dative]. The thing you like more is in the accusative.
Er zieht das Landleben der Stadt vor.
He prefers country life to the city.
In the meaning 'to move something to an earlier time', the verb is often combined with time-related nouns such as 'Termin' (appointment), 'Treffen' (meeting), 'Besprechung' (meeting/discussion), or 'Reise' (trip).
Wir haben unseren Urlaub um eine Woche vorgezogen.
We moved our vacation up by one week.
The literal meaning 'to pull forward' is used almost exclusively in connection with curtains or drapes, in order to cover a window.
Es wird dunkel, du kannst schon mal die Vorhänge vorziehen.
It's getting dark; you can go ahead and draw the curtains.
This meaning is a technical term from gardening. It is used for plants, herbs, or vegetables that you grow indoors from seeds before planting them outside.
Ich ziehe meine Tomaten jedes Jahr auf der Fensterbank vor.
I start my tomatoes indoors on the windowsill every year.
to move / lay / misplace
'verlegen' and 'vorziehen' can both be used with appointments and events. 'verlegen' is more general: an appointment can be moved to another day, another time, or another place. 'vorziehen' is more specific: the appointment takes place earlier than planned. The words can be confused because 'vorziehen' can be one type of 'verlegen'. If it is only clear that the time or place is changing, you say 'verlegen'. If something is explicitly supposed to take place earlier, 'vorziehen' is more precise.

Am Montag zieht Samir die Besprechung auf neun Uhr vor.
On Monday, Samir moves the meeting up to nine o'clock.
Wegen des Streiks verlegt die Schule die Prüfung auf nächste Woche.
Because of the strike, the school is moving the exam to next week.