Make another branch as master

Since the task was to simply use another branch instead of master, you can simply remove master branch completely or rename it to let's say - legacy, then take another branch and rename it to master. That's it. Here are actual commands that you might need to execute to achieve the goal locally and on GitHub:

git branch -m master legacy               # rename local master to legacy
git checkout legacy
git branch -m another_branch master       # another_branch will be our new master

Locally we are done now. However you cannot simply remove master branch on GitHub. You need to take another branch as default first. This can be done in repository Settings > Default Branch. Once you do this, you can proceed:

git push origin :master                   # remove master on GitHub
git push origin master                    # push out our new master branch
git push origin legacy                    # push our legacy branch too

Then go back to Settings > Default Branch and switch default branch back to master. Additionally you can remove all extra branches that you might have created during migration process.

Alternatively, if you are looking to save all your actions in history, check a correct answer here.