How much Analog would a Beginner Electronics person need to know?
When it comes to electronics, there is no such thing as pure digital.  What I mean by that is a "digital signal" is still an analog signal at the most fundamental level.  Only in very basic systems can you ignore the analog-ness of those digital signals.

Here's a short list of things that a "digital guy" needs to know about analog stuff.  This isn't a complete list, but is a start.  They are roughly in decreasing order of importance:


Ohms Law.  Super important.  Related:  know resistors inside and out.
Know the general properties of caps and inductors.  You don't need to know every property or every formula, but a general knowledge is good.
Know diodes.  The important things to understand are breakdown voltage, forward voltage, and power ratings.  
Have a passing knowledge of signal integrity.  This includes trace/wire impedance, signal termination, decoupling caps, AC signal return paths, and overshoot/undershoot.  Again, you don't need to know the formulas by memory, but have a passing understanding of the things involved.
Know linear voltage regulators.  You need to understand things like dropout voltage, converter efficiency, and power dissipation
Have a passing understanding of a switching regulator.  Have a passing understanding of the standard buck and boost topologies.  
Know various digital signaling standards.  Things like TTL and CMOS signaling at a variety of voltages.  Open-collector signals.  Differential signals like LVDS.  Etc.
Know how to read datasheets, especially the stuff about power levels and signal levels.


For #2, 4, 6, 7 the goal of a "passing understanding" is to know enough to be able to Google for more information when the situation comes up.  Meaning that you know the terms and roughly  where this stuff is used enough to know when and what to Google for.  

For the others you should know quite a bit before you need to Google.  

I don't have a book to recommend (every one I've read has been terrible).  Sorry.

Edit:  Someone edited my answer to say along with #1, "Also Kirchhoff's Laws: KVL (Kirchhoff's Voltage Law) and KCL (Kirchhoff's Current Law)."  I actually disagree with this slightly.  This falls into the category of you need a passing knowledge so you can Google it if it becomes important.  I've worked professionally as an Electrical Engineer for 20+ years and I've only had to use the voltage law once, and never the current law.  I un-edited my post to remove this.