I'm going to add a third option, a phrasal verb, "to look for." So we have: "To look for," "To seek," "To strive."
"To look for" is ordinary, everyday language. It means I am trying to find something. It means I am making a search.
"I'm looking for my notebook. I can't find it. Has anybody seen it?"
"To seek" means the same thing as "to look for," but it is more formal and literary. It is often used when you are trying to reach some important goal. If I were puttering around trying to find my pen, my wife would never say "What are you seeking?" She would say "What are you looking for?"
"I'm looking for my notebook."
"I seek beauty and excitement in my life."
In a Monty Python movie, the guard at the bridge asks "What is your quest?" and the knight replies "To seek the Holy Grail."
"Look for" and "seek" emphasize the thing you want to find.
"To strive" emphasizes the hard work you are doing and the effort you are putting in.
"Seek" is transitive. It takes an object. can say "I seek [something]." "I seek adventure." "I seek the magic key to learning English."
"Strive" is intransitive. You can't say "I strive [something]." You must say "I strive to do something" or "I strive for something." "I strive to be a good citizen." "I strive to do the best I can."
My I ask: by any chance, have you been reading Tennyson's poem, "Ulysses," which closes with the words:
We are not now that strength which in old days
Moved earth and heaven, that which we are, we are;
One equal temper of heroic hearts,
Made weak by time and fate, but strong in will
To strive, to seek, to find, and not to yield.