In most software systems, it is a good idea to put the version number in a place where the program itself can display it to anyone who needs to know it. That requires adding it to a code or resource file. Where exactly depends on the technology involved, and who the persons are who need to get informed.
For example, it may be the end user who shall be able to find out the number easily, so you need a function to display it to him. This way, a support hotline can ask users which version they currently use. Or it maybe just the administrative staff. Or in case there is only one installation of your web app, you can decide to keep the version number fully internal, since "support" may be able to check this one unique installation directly.
If your "readme" serves as changelog, then yes, the version number will also occur there (obviously, how would you otherwise make references to change descriptions)?
I have no idea how to version the branches
New version numbers are assigned for each release of your software, to "production" and/or "test" environment. You should not release from arbitrary branches, you release only from your "Master" branch or a special "Release" branch, from nowhere else. And if you have bugfix branches or feature branches in use, you either integrate them before deploying a release into "Release", or not.
Therefore, it is usually only important to mark the version in the repo inside the "Release" branch, exactly at the point in time when a new release is created. A standard tool for this are tags, but you may decide that for test releases a version number in a commit message may be enough. That is up to you and depends a lot on how your team worksin general and the kind of system you build.
So from an outside point of view, you create the illusion of changes applied to your system in a linear order, one after another, each group of changes with a higher version number. The exact version number which is written into the code files of a certain branch is usually unimportant and should not bother you.