Here is my advice:
Hakko FX-888D + Chisel Tip - This makes up more than half of your budget but if you're comfortable with that, it's a great iron. Ordering from TEquipment instead, as nanofrog suggested, makes a lot of sense. You'll also need some solder(the little tube in the roll-up kit doesn't count

) and good solder is expensive.
Sparkfun Multimeter - It's hard to give any advice on this unless you're willing to buy a more expensive meter. But no auto-ranging, doesn't power down on it's own, etc. Also, generally cheap meters like this have pretty awful continuity testers, but this is just a hunch.
Six 25' Spools of 22AWG Wire - This seems like a good price especially considering you get six colours and a box that dispenses them.
Aligator Clip Leads - Good price but I'd think if I had to choose between the two, I would opt for female-male and female-female test leads instead.
Roll-up Tool Bag Kit - Convenient. But I think that you could source the components more cheaply. Also, the look of the solder, wire-strippers, cross-lock tweezers, and screwdriver set don't impress me.
Screwdriver Set - Cool.
But one thing I'm wondering about this list: Is this just the
equipment you're buying? Because you have no parts to play with. Sparkfun will have competitive prices on genuine Arduino's, etc if you want to throw some stuff like that in too. Their components are expensive though.
Lastly, do you have a breadboard? Some sort of power supply? Small collection of passives? Because you should get all of those things too

You might be interested in getting something like
this or
this so that you don't have to worry about resistor/capacitor values for a while.
Maybe we can have a follow-up digikey cart when you have some new projects in mind
