I received an overall better job offer one week after I started to work for another company. I feel like there is no point in mentioning the 2 week leave notice as I don't have anyone to train in any task only I know how to do, and there is nothing to finish. How I should proceed? I'm aware that most likely I will burn some bridges, but I'm willing to take that risk.
Talk to your manager before writing a formal letter. There's a pretty good chance they'll agree to an immediate exit since they won't want to waste two weeks' salary for you to twiddle your thumbs, especially since you've only been there a week.
If you have started some work/task, be prepared to be asked to stay a short time to complete it.
I once had a developer ask to leave after only a month into her probation. We agreed she would stay to the end of the week to complete her latest piece of work and tidy up some documentation. While I was a little annoyed, there wasn't any hard feelings as I knew the other job was actually a better fit for her.
However, you should also be prepared for your current employer to be upset - filling a job is time consuming. They may go so far as to put you on a do-not-hire list, or even be petty enough to force two weeks out of you (highly unlikely though).
Depending on your location, you might even get away with simply walking off the job - but that will certainly burn bridges.
If you work an office job, it is likely that for legal purposes, the company will ask for a resignation letter no matter what. They can deliver your last paycheck on time and do any de-briefing that needs to take place.
If you don't work an office job, chat with your manager. There is likely to be some level of distress, so anticipate that, keep a cool head, and don't do anything that will wreck your chances of making a clean exit.