As I've gone along, I've found that there is a list of items I always have on hand when making string art myself. I keep all these in a box, so I can pick it up and take it with me when I want (and it stops me putting bits down and loosing them).
The pieces that I make generally use sequin pins, which are very small and can be placed tightly together. This little whale for example is 10cm long from the end of his nose to the tip of his tail and there are about 3 pins every centimeter. With such a small space between pins, wrapping the thread around them and pushing the thread down are both difficult. Some of the tools I use are especially for working with pins, but could be used for any type of string art.
Here is my list of essential items for making string art (in no particular order).
1. Pliers - when I make my bases, I pre-drill the holes and add in the nails or pins afterwards, hammering them in to the same height. But with sequin pins a hammer is a bit much, especially when the pins are close together. I hold the pin lengthwise using the pliers and gently push them into the holes.
These are perfect when I bend pins and have to pull them out again. I also find them useful sometimes for pushing down the thread.
2. Lolly Sticks - useful for pushing down the thread. I keep a couple of sizes on hand.
3. Masking Tape - for taping the template to the base when I'm making guide holes with...
4. An Awl (the blue pointy thing on the left) - I tape the template on and then poke holes in it. This shows me where to drill the holes and also helps me get the same arrangement of holes on each kit. Can also be used for pushing down thread.
This is a bit too big for poking guide holes for sequin pins, so I use ...
5. A pair of Compasses - I use my old school compass (pair of compasses?) to poke the guide holes for sequin pins.
6. A circle cutter - originally purchased to cut circles in felt, I sometimes use this to poke the guide holes for sequin pins.