How do officers work in promotions and which section (Military Intelligence, Signal, Infantry, etc.) would be best for promotion?

A few thoughts:
- You are asking the wrong question. You ask what branch is best for promotion when you should be asking what branch is best for you.
- To answer the above question, ask yourself what are your personal and professional goals over the next 1, 5, and 10 years.
- Then research what branch will best help you to reach your personal and professional goals. Consider both during military service and after military service.
- All branches have same basic requirements for their officers but each branch has additional specific requirements for their officers. For example, if you hate numbers then I recommend you avoid the technical branches (Engineers, Field Artillery, CBRN). If you want more opportunity for special schools (Ranger, Pathfinder, Airborne) then consider combat branches over CS or CSS branches.
- At the end of the day, you will be more competitive for promotion (regardless of branch) if you are in a branch that you love rather than a branch you want to be in that is "best" for promotion. "Best" for promotion will most likely change in the future by the time you reach that gate anyway.

I recommend you download and read AR 600-3, which is a short-ish Army Regulation that governs the promotions and career tracks of officers and is broken down by branch. I recommend you read chapters 1-6, then read the branch chapters that interest you. The part in the branch chapters you should focus on are the jobs and schools the chapter outline as key and developmental ((KD, required for promotion), as well as recommended broadening assignments in between KD jobs.
This is important because you can essentially look at the roadmap per branch and picture yourself in this career field at the different ranks. Don't focus so much on the LT years, but on the captain and major years.
You should listen to the O6 above, his advice is spot on. The information he you should base your decision on is in AR 600-3.
Promotions are based on two things, authorized MTO&E positions across the Army per branch, and at large bids. So, hypothetically, if the Field Artillery (FA) has 100 authorized MTO&E authorizations for majors in the Army and Joint force, and 80 FA Majors, than they will select at a minimum 20 captains to promote to major. Them the at large bids are branch immaterial meaning any branch within the Operations Category, formerly known as combat arms, may compete for these bids.
Let me back up a step. During the promotion board, all operations officer board files are viewed by every member of the board and scored. The computer system orders all files based on the scores to generate an order if merit list. So all officers, irregardless of branch within the operations category and stacked into the order of merit list. The, starting at the top, the assign each member of a branch one of the authorization slots available within their branch. So the highest guy in the FA branch gets the first of the twenty FA authorizations. Once all the authorizations are gone, then the board takes the at large bids, and starting at the top of the OML, goes down the list and gives them to those officers highest on the OML but not assigned one of their branch authorizations. Once those are gone, the board ends. Those given their authorizations within their branch or at large bids are promoted, and the rest are not that FY.