I would advise using git to manage future updates.
Make a clone of the main git repo:
git clone git@github.com:moodle/moodle.git
Then create a new branch based on the version of Moodle that your site was based on (tagged v2.7.3):
git checkout v2.7.3
git checkout -b [name of branch to create]
Once that is done, copy your current site's code over the top of this. Now look through and commit each of the custom changes (tools such as gitk or IDEs, such as PHPStorm, with built-in git support can help with this).
Once all the changes have been committed in sensible chunks, it is time to upgrade to the latest Moodle 2.7, by creating another new branch and rebasing it on top of the latest Moodle code:
git checkout MOODLE_27_STABLE
git pull
git checkout [name of branch created above]
git checkout -b [a new branch name for your upgraded code]
git rebase -i MOODLE_27_STABLE
This may work cleanly, or you may need to resolve some conflicts as you go along (depending on what was changed in the original and what changes have happened in Moodle).
Any future upgrades (even between major versions) can be completed using the last set of git commands (but, when upgrading between major versions, you will need to trim out any core Moodle commits during the 'rebase' step).
I would also advise coming up with some sort of consistent naming convention for your site branches (e.g. based on the Moodle version number).
