Most of the way through the list and the single highest cost is the $5 “handling fee” from digikey for having an order below $25
If I can substitute Mouser parts for Digikey, I can substitute any brand with the same values? I’ve got bunches of old resistors and capacitors. If they need to be new, can I get ones at Radioshack to save on shipping?
I have pill bottles that should be the right size. I have the battery hookup. Saves me $3.50
For what it’s worth, I was in this same boat about a year ago, and now I feel like I’m up to the point that I wanted to be at and am even going further.
Here’s what I did:
I got one of those 300 Projects in one! Project Kits from Radio Shack. If you get that model linked to or better, it’s actually practical knowledge that you can transfer to later projects, because it has an actual breadboard in the middle of it and it comes with a bag of IC’s, a bunch of resistors, transistors, capacitors, etc. that you plug into it, so you’ll get used to using the actual parts that you’ll use later on in “real” (from scratch) projects. The manual in the Radio Shack kit is actually written by Forrest Mims III (see #2), and is amazingly informative. I felt like I was actually learning what I was doing instead of just learning how to make the projects.
I got Getting Started in Electronics by the aforementioned Forrest Mims III. He’s a great writer and is specifically adept at teaching the concepts of electronics to beginners. This book acted as a great supplement to the manual that came with the project kit and expanded on things that I was curious about.
Seriously, you could do a lot worse than to just get those two things and work through a bunch of breadboard projects.
Piggybacking on engineer_comp_geek’s suggestion here:
Visit RadShack (or any local electronics supply house) and get a spool of desoldering braid. To use it, simply unwind a bit and use the tip of the soldering iron to press it against the joint you want to desolder. As soon as the joint heats up enough you’ll see the braid wick up the solder (sluurrrrp!), nice and neat. Remove the heat as soon as you see it happen and you’ll save yourself headaches due to running solder or too much heat.
LED projects are fun ‘n’ easy. For example, a real easy project that you can do with parts you may already have on hand is to add an LED to your computer mouse – with a bit of creativity, it can be VERY cool (Google it, or look here for an example.) LEDs are useful because they are reliable, they only work in one ‘direction’ (so you know right away if you mess up), they’re low-voltage, and you can pry them out of just about every bit of junk out there (old VCR’s are a great source) for nearly free, or purchase them for dirt-cheap as well.
Pick something small that you can build on as your first ‘real’ project. For example, my son decided to make a cat toy as his first project. We picked up one of those bubble balls that those cheap toys come in, scrounged a tiny motor from an old VCR, and dug up a few other bits (resistors, wire, LEDs). Total cost was maybe $8, and most of that was the battery. Our cats have a blinky-light ball that flops around on the ground to chase around now. One of them loves it, the other hisses and runs, but hey. 
Anyway, the point is, my son learned how to wire up an LED first, then how to wire up a switch, then a motor, and so on.
You can use old resistors, but be careful with old capacitors. Certain types dry out when they get old and they don’t work so well. There’s nothing particularly fancy in your circuit, so you can probably buy a lot of the parts at Radio Shack. They won’t have the transducer or the coil, and I doubt that they’ll even have the counter chip. But the LM386 is a generic amplifier chip originally designed to power headphones in walkman type thingies (for us old farts who remember what a walkman is) and I’m pretty sure Radio Shack carries it.
You have to be careful about substituting certain things. For example, don’t substitute a mylar cap for a ceramic disk cap of the same value unless you are certain that the circuit doesn’t depend on some property of the ceramic cap. Generally speaking, you can substitute one IC for another of the same IC made by a different manufacturer. The LM386, for example, is made by several folks, as are a lot of other chips. Sometimes though you run into something weird which forces you to use a particular manufacturer’s chip, and the circuit won’t work right when you stick another manufacturer’s version in there. For most hobbyist type stuff, you aren’t going to run into this sort of thing very often.
Can’t I use my multimeter to be sure that the capacitors are still good?
And can’t I get a ferrite shielded tuning core from an old TV or radio?
I forgot to ask. Is there anything I might have that uses the binary counter chip? A computer part, or handheld game, or remote control or something?
You can use a multimeter to test for shorts and if your meter has the capability, you can measure the capacitance as well. It won’t tell you if the capacitor is leaky and it won’t tell you if the capacitor’s working voltage is compromised.
Based on how this part is used, I would recommend just ordering the part for this one. You’re messing around with a tuned circuit.
These days it’s not likely. Most of the things that might need a binary counter have a microcontroller or an FPGA already in them, so the binary count function can be easily programmed into an existing part instead of using a separate chip to do the job.
Just FYI - Snap Circuits are teh awesome! 
I can make exciting space sounds and have the little fan thing jump up in the air.