search for sth/sb
phr. v..phr. v.. to look carefully for someone or something that is lost or hidden.
phr. v.. to examine a place or area systematically in order to find a specific person or object; often implies a more thorough or formal effort than 'look for'.
I search for my keys every morning before work.
The rescue team continues to search for survivors in the mountains after the storm.
Historians often search for primary sources in dusty archives to uncover the truth about forgotten events.
the object is the thing you want to find, while the place you look is the direct object of 'search' alone.
contrast 'search for something' (the goal) with 'search a place' (the location); for example, you search a room for your phone.
I searched my phone everywhere.I searched for my phone everywhere.without 'for', it means you are looking inside the phone's software or hardware, not trying to find the device itself.