If you can afford it, upgrade.
I have an EXtech EX330 multimeter, lady ada sells that also. Got it on sale for $40.
I have a 60 W Weller soldering station, its moduler with really nice flexible burnproof cord, is similar to the Hakko 936 lady ada carries. Not a model Weller makes anymore. No adjustments, just change the tip used if I want more heat. Have had the same tip in it for years, small enough to solder SMT parts, big enough to heat larger pads for removing parts.
I have a solder sucker, nice for cleaning up solder blobs, or for taking parts off boards.
Small size Solderwick, like 0.075", for cleaning up excess solder that the sucker can't get, like off legs that won't fit in a socket or thru a hole.
I have a panavise, much better than the alligator clip things (I don't find those good for much besides holding wires to tin the ends, they don't seem to stay in place well).
I have a handheld lighted magnifying glass from Great Point Light, good for checking out solder joints up close. (rectangular, ~4.5" x 1/75", LED lights up when you slide it open - look in walgreens, walmart).
Other tools - I would get a multisize wirestripper, easier to strip stranded wire more accurately (and not lose so many strands).
Needle nose pliers - small ones are nice, I have several different sizes.
Small cutters - good for trimming component leads & wires - I have several.
Solder - I just got some 23 guage. 0.025" diameter - really easy to get just the right amount of solder and not have to clean it up later.
Wire - purchase as needed for your project.
Breadboard - I would get 2-3 830 point boards, lets you experiment with more components, can spread things out a little more to make it easier to see what you're doing.
Terminated wires from pololu.com and crimp connector housings - these are really nice for making up cables to connect your stuff.
Jumper wire kit - I bought a couple bags of 50 wires in different lengths for breadboarding. I have used 24 guage solid wire over the years, just stripping as needed (basically just telephone wire). The flexible wire with the little pins are much nicer to use. Not sure where I got them from, but its this kind of thing:
WIRES-PACK-M/M Jumper Wires Pack
After that, parts as needed for whatever you're designing.
I wirewrap a lot of things after prototyping sections on breadboard, beats point to point soldering.
So sockets to support that, and some boards like Vellerman ECS 1/2 to build up on.
So socket strips like these
http://www.marcospecialties.com/product.asp?ic=SIP20L
or similar.
I've also bought resistors in bulk packages - something like 8-10 each of a mess of values, for $10 kind of thing.
I have done PCBs via expresspcb.com a couple of times, I find their software easy to use. Just check the size of thru holes on things like SIP sockets, make sure they are what you need. Very easy to pick something undersized.
You are not far from pololu.com in AZ, go poke around their site too.