Sounds like an excellent start.
With these breadboards you often need to link things together with wire (say to jumper pins together, or connect power to something).
My advice would be to go out and get one metre of cat 5 cable (network cable with solid core wire, not the "flexible" stuff) That will have 8 wires in it. Cut them as you need them into short lengths (could be a couple of cm). That wire is well suited to these breadboards, and ample for the very small currents you're likely to be using.
I would advise that you get a battery holder that holds either 4 or 6 AA cells as these will be cheaper in the long run, and supply more current than a 9V battery.
If you get hold of some CMOS chips, or MOSFETs, remember that they are static sensitive and should be stored and handled appropriately. Generally they will be transported in some suitable packaging, so keep the bag or tube to store them in after use. A suitable alternative is to cover wrap some expanded polystyrene in a couple of layers of aluminium foil and stick the components into this.
When you handle static sensitive stuff, try not to touch the metal pins and work on a surface that doesn't hold a charge (i.e. plastic is bad, paper is OK, special conductive mats are best, but probably not worth the expense at this stage). If you take basic precautions you'll be pretty safe.
If I were you, I'd grab one or two LM555's and a couple of 4017's and maybe a couple more LEDs -- it will probably set you back a couple of dollars. You have 9 LEDs in that kit, but you may want to monitor a few more places. Oh, I'd get a few more diodes (1N4148) and general purpose NPN transistors (2N3904, 2N2222, BC548, etc) -- you're likely to use them for the circuit you want to make. Oh, and you're driving an inductive load, so grab 10 1N4001 diodes.
You can get much of that from Nightfire too. Just make sure that you don't order SMT parts (for ICs they should be DIP-xx, not SOIC-xx)
(I promise I can't think of anything else). None of those components are critical, so if your supplier doesn't have exactly that component, then something similar is just as good.
In most cases you can "cheat" and run LEDs directly from the outputs of CMOS. In other cases where you need resistors, things are often not really critical, so if you run out of one value, use the closest one you have.
If you have any questions feel free to ask.