Core updates and module updates: Does order matter? I inherited a badly out-of-date Drupal 6 multi-site instance.  There is only the production instance. (yeah, that gives me a warm-fuzzy!)

Many modules are several versions behind and the core is several versions behind.

Does the order in which I perform the updates matter?  I'm thinking update core first, then update the modules, but I'm just not sure.

Anyone care to advise?
Order does matter.

Sadly, it's hard to tell exactly how it will matter in your situation. If you can, make a dev copy of this site and experiment on it. I would either go with core-first or with all at once. But that's my intuition, and actual tests on a copy of actual site matters much more.

Usually it's OK to jump all versions in-between the one you have and most recent one. For modules, sometimes, rarely, it's good to use most-recent minor version and don't install next major version at all (if both are maintained), or install most-recent minor and then update major. Sadly, it's different for each module, so fastest thing would be to update everything straight to the recent version, and see if / what broke. Of course, this is only feasible on a dev copy, not production.