I'm running version 2.0.6 according to my admin panels. Can I go straight to the most recent core upgrade or is there any reason to go via the intermediate versions?

Heres what you should do.
Save any files you have in the modules dir, upload dir, your mainfile.php and themes.

Then download 2.0.13.1 (full, not upgrade)
Take out the modules dir, and save it some place
put those files you saved (modules dir, upload dir, your mainfile.php and themes) in, in the correct places of course.
Then put the modules dir back in. If you are using Newbb2, then you will need to remove the newbb dir before putting the modules folder back in and also for the news module as well.

Thanks for the advice!