Manually update core from 7.0 to 7.78 in one fell swoop
The latest update contains all of the updates before so going straight to the latest one should be okay and is what is usually recommended. Be sure to backup your site and test the backup on a dev copy.
You should update your contributed modules at the same time as some of them may rely on updated versions of Drupal (core).
Lastly and very importantly, there have been numerous security releases since the 1st release and some of them are quite bad so I would say you should probably assume your site is compromised. That's 78(!) releases you've missed out on, including the infamous Drupalgeddon.