They are roughly equivalent, but there are subtle differences. All express the idea of searching, but "Seeking" implies desire. You're not just looking or searching - you are also wanting. "Search for" and "look for" are very close, but "search" implies more effort. You can walk down the street looking for a Chinese restaurant. If you find one, great! If not, so what?You can go to a different kind of restaurant. However, if you are searching for a Chinese restaurant, you are more likely not to accept anything else.

'Seek' is more literary, antiquated, or used in some common expressions. It suggests the seeker is on a quest or on an important undertaking. 'Seek and you shall find,' 'I hope you find what you seek.' 'What do you seek?' Not commonly used in normal speech (US)... sounds like something from the Bible or Lord of the Rings. Although 'seeking the truth,' is a commonly used expression. To 'look for' is more general and common in speech. You could be looking for a new job, your keys, your friend, etc. To 'search for,' is similar to 'look for,' although it could imply greater importance or a concerted effort to find something. It sounds like you may be trying harder or for longer when compared to 'look for.' Sounds like a more serious pursuit. You also say 'I'm trying to find...'