Look for is the standard verb (in this case, phrasal verb), meaning to try to find.

Search is:

1) for something to look for heavily. I searched my house high and low, but could not find my passport. 2) to try to find on the Internet (search the Internet is a collocation, meaning a common word combination).

There is a third word, seek, which is used in certain expressions & tends to be a bit more formal. Ex. seek justice, shelter. Besides, sometimes you see it on car radios, where it is used to mean find the station that you want.