Which section to start with when writing a research paper

There's no 'right' order. Starting where you feel able to do so is far better than not getting started. I usually write lots of sections concurrently, or start writing and work out what the sections are later.

On the other hand, it will make your life somewhat easier if you can work roughly in order of dependencies, so you don't have to keep changing what you wrote earlier. From example, it might be helpful to write out some of your notation before you start using it. The introduction will often come late in the process for this reason.

This almost exclusively depends on the personal taste and habits of you and your co-authors.

There are different types of writers and different types of projects. The order of writing has to be suitable for these. If you and your co-authors are absolutely certain that what you did already makes a nice and complete paper story, then writing the methodology and results first makes sense - you can adapt the introduction accordingly then. However, if the final scope of the paper is not 100% fixed already, writing the introduction first makes sense, so you can check whether your project-so-far actually reads complete. The literature review may then have some impact on your methodology section, so that you can make the distinction to previous work very clear.

Edit-as-you-go style writers may also want to write the introduction first so that what is written so far is always in a clean state. For others ("binge writers" and writers that iteratively refine detailed plans) it may not really matter again, so the properties of the project can dictate the order of writing.

There is a plethora of literature on successful academic writing, and different books will advise different approaches, so there is probably no unique answer to the question.

I would just like to address this comment from the TC:

Everything is OK, I just heard one conversation that without a written hypothesis and proper literature review no one should start to write an article. I thing that this part belongs to the Introduction but it is possible to have it only in mind when I actually writing a paper and put it on it later. I´ve been just curious, how other people writing their papers.

The order Method > Results > Discussion > Introduction (let's call this the MRDI method) is extremely common in my field (biomedical). From what I can tell there is nothing wrong with it, though your comments are certainly worth discussing.

It's important to distinguish between conceptualizing and writing

Generally, the path from concept to manuscript is not linear or singular. Some phenomenon, questions, or challenges might have sparked us to gather some evidence and perform a hypothesis test. At that stage we might not have writing an introduction, but we would have already known the big three in the introduction section: what is known, what is not, and how can our work help here. So, in a way, your quoted comment seems to have mixed up "research" and "writing." Yes, we may not have the introduction written first, but the big picture is already conceptualized, and recorded in some form not necessarily as a manuscript.

Hypothesis integrated analysis plan

As for the comment on written hypothesis. Again, I think the speaker mixed up "research" and "writing" or he/she was speaking to a group of very new science students. In my field, all analyses were based on hypotheses, and all procedures are predetermined. In a sense, if we come up with any results, it's already implied that hypothesis setting and analysis planning have already been completed.

Don't fuss over the real "order"

Personal experience: Don't sit down and think "Okay, I am going to write the Discussion section and nothing else!" When write, just let your thoughts flow, if later the materials deem more appropriate to be moved to Introduction, so be it.

Same goes for Results section. Some people were very disciplinary about not to interpret any findings in the Results section. But I just write whatever I want in the draft, and then parse out the interpretive parts later. For me, thoughts are thoughts, they just come out like a chain of... sausages of different stuffing, for the lack of a better analogue. After I am done, I'll then go back to cut them up and categorize them. As time goes on I am getting better at churning out sausages of similar stuffing in one chain, but I have no desire to compartment my thought process like how writers compartment a research article.