You didn't specify the types of robots you are interested in, so please excuse the following general ramblings:
Consider Lego or Meccano for building smaller robots. The reason is that the hard work of producing standard size and shaped parts with accurate dimensions has already been done for you, along with gearing and connecting them to motors. The interfacing to an Arduino is relatively straightforward, and well documented, without the need for an expensive controller like Lego Mindstorms.
Adafruit's basic tool kit (as linked to in the post above) will serve you well on the electronics side. If you are going to be harvesting components from other electronics, some desoldering braid and a solder sucker will come in handy.
For a general set of tools, consider the following:
Birthdays and Christmas are great times to acquire tools. Dads love to buy tools, and Mothers love to buy safety equipment.
The following tools are not optional, in my opinion:
o eye protection i.e. safety glasses. A small piece of wire hitting you in the eye while you are trimming components is not good.
o a small fan to blow away the fumes while you are soldering. Mount it to the side, so it blows across your work area at low speed. You can get fancier setups to handle fumes, but this is the minimum.
o a small first aid kit You will inevitably burn yourself on a soldering iron, or cut yourself on something.
o a small fire extinguisher If you are using bigger battery packs for robotics, (and pushing the envelope
) overheated wiring or a short are bound to happen and a fire is inevitable. Be prepared.
For hand tools:
You can pick up a basic set of tools from Costco or elsewhere as an All-In-One kit, for about $50. These are of modest quality, but you can replace them as they break or wear out with better quality tools, and as you figure out your needs. The kit includes screwdrivers, wrenches, pliers, wire cutters, tape measure, hex and torx wrenches etc. You can take pretty much anything apart and put it back together with one of these. These will serve you both for robotics, and in general use for the rest of your life.
If you don't have a dedicated work area with lots of storage, buy two tool boxes. A medium sized one (about 20-24") for the tools you use all the time, and a big, deep one (about 30") for the tools you don't need that often. Toss the display case that came with the original set of tools from Costco.
You can augment the above kit over time and as you need them with the following:
o Jewellers screwdrivers for tiny screws (these may have been included in the above kit)
o small-sized pliers and wire cutters (handy, but not needed)
o assorted screwdriver bits (I have a collected about 100 over the years), for all the odd screw types like security bits, triangle, pozidrive etc.
o hacksaw and/or wood saw, small enough to fit in a tool box
o a set of exacto knives for precision cutting
o other specialized tools as needed (chip extractors, oscilloscope, digital logic probe, hole punches, nibblers etc.)
For power tools:
o 3/8" variable speed drill (cordless is nice, but not necessary, and a bit of a pain if you have to wait for the batteries to charge)
o drill bits (high speed steel for wood, aluminum or thin steel, cobalt for thick steel)
o a small drill stand for the drill (optional)
o Dremel kit or similar knockoff (optional) Do keep this display case, as you won't use the Dremel that often and it has lots of tiny bits.
o heat gun (optional) Useful for shrink wrap tubing, and shrinking pop bottles if you make your own battery packs.
o wire in assorted colors and gauges
o nuts, bolts, washers etc. Available in kits with various sizes.
o solder, glue, electrical tape etc...
o rechargeable batteries are cheaper in the long run
I hope this was helpful.