That's a nice set. I've knitted that small, my basic advice is to use needles about the same size as the doll's fingers or their wig size. I'd tell people wanting to start to do the same thing, choose needles about as thick as your fingers. If you want/need a loom then a sock loom is good for dolls.
I need to work on the not buying so much end of things. Getting to the point where I'm ending up with doubles of things I don't need doubles of. On the other hand I do have a $25 limit gift (it was on sale) if I find myself in that situation.