I don't know, but something wrong with both for me. I feel like "Let's go find my bike".

The first one is good. You can also say "Let's go look for my bike." "Let's go find my bike." "Let's go see if we can find my bike." "Seek" is usually looking for more important things, things that aren't concrete: seek advice, refuge, your fortune. The noun after it is the direct object, so there is no preposition.

I would say either, let's look for my bike or let's search for my bike. To include the word 'go' in the sentence I would add 'and'. Let's go and look for my bike or let's go and search for my bike.

Your last sentence is wrong, however you can say "let's go searching/to search for my bike".

"Let's look for my bike," and "let's go look for my bike," sound the most natural to me. I agree with Goatish about the word 'seek'.