Found a job, signed the contract and today unexpectedly got a better offer from another company?

Hello everyone

So as the title says, I got a job offer from a company where I am not thrilled to work at but anyway signed the contract and am supposed to start beginning of January.

Today unexpectedly I got an offer from a company where I would actually be happy to work, the salary is the same as the other offer, but the project, people and culture seem MUCH better suited for me. I told the second company I can start work in middle of January.

What do I do? Do I go in first day of January at the first company, directly hand in my 1 week notice and THEN sign the second contract or should I notify them right away today? And I guess, until I resign from the first company, I can't sign the contract of the second one, right?