Where can I find a weekly source of world news that's comprehensive --
it should not assume that I have familiarity with stories from other
news sources?

I want something:
- objective - minimal political or theological slant, no editorials
- well-digested - no breaking stories with incomplete information
- US-centric or strictly international

I'm sick of the US and European press and its editorials masquerading
as news, as well as the hype designed to drive the daily news cycle
even when there's little notable news. I want to stay informed, but I
hate being insulted by today's printed tripe.

I suggest The Economist.  It's what the world's CEOs and leaders read.

They really describe everything that goes on in the world in a very
simple and explanatory way, while giving great analysis.

However-- while they show both sides of an issue, they will add in
their own views. For instance, writing about the genocide in Darfur,
they would say that Sudan's leader should be stopped and China
shouldn't allow them to get away with it (because China buys huge
amounts of oil from Sudan and doesn't say that the genocide is wrong).

Since their views are for the most part enlightened and correct,
though, this is not a problem for the world leaders reading it :)
It's the most complete source of international news that you will
find, I guarantee.
It's great for both US and international news.
Bill Gates says he reads every page, if that matters to you. 

Anyway, it's great and I highly recommend it!