Personal electronics tool kit
Most important tool: 


A quality soldering iron.  Mine's a Weller.  Those Radio Shack pencil irons are useless, and will only make life more difficult.  
A wet sponge for cleaning the iron tip, several tips for the iron.  
Both desoldering braid and a solder sucker.  
Plenty of 60/40 solder, the silver bearing stuff doesn't flow nearly as well.  A well stocked junkbox is extremely important.  The more components you have to hand, the faster you can work.
A good digital multimeter, I love my Fluke.  Add to that a good analog multimeter, digital meters offer no sense of change over time.  
A dual channel 100MHz oscilloscope (mine's a tektronix mainframe that I scavenged).  
Good lighting!  A third-hand rig or similar to hold parts and boards that are being soldered.  
A spectrum analyzer is very useful for radio work.  
A function generator.  
A sharp pair of flush cutting nippers. 
Several spools of different coloured 20 gauge hookup wire.  
Some people keep flux to hand, but I've never found I needed it.  
Assorted scraps of printed circuit board, or strip board - breadboards are very finicky and can add circuit-killing parasitic capacitance.  
The absolute best tool I've found for cutting PCB is a jeweler's saw.  You can buy them from Contenti, and aren't very expensive.  I've never seen anything use produce a better or faster cut.


I think that covers the bulk of it, it may seem like a lot of stuff but it's mostly small stuff.  An LCR meter is handy, too.  I have a foredom flexshaft, it's kind of like a Dremel but more powerful and higher quality. They're not cheap machines, but are a real life saver for modifying circuit boards.