Updating from an ancient system

Does anyone have any idea what the safest way to go about updating (from a very very old system dating back to 2011) is? I had been updating through my (GoDaddy) FTP as I once lost my whole blog updating through wordpress (and was given a stern warning by GoDaddy to only update through their web based FTP).

But it doesn’t seem to be working – i.e. when I hit the update button it doesn’t successfully make the ‘leap’ from 3.3 to 3.8.

Would it, in this case, be better to go through WordPress directly to automatically make the update? Or is that dodgy? Should I instead upgrade each version manually (labour-intensive but maybe the only ‘safe’ way)?

Please help… I’m a bit lost..