Lesser Goldfinches (Carduelis psaltria) are common here in Tucson, and these small, lively, yellow birds are easy to attract to your yard with special goldfinch feeders filled with thistle seed.
Goldfinch feeders are elongated tubes with a series of tiny seed holes and short perches that keep larger birds (like greedy doves) from eating the relatively expensive thistle seed. The best goldfinch feeder I've seen is an elongated tube with sides made of fine-mesh metal screen. The tiny thistle seeds poke through the screen, and only small, light birds like goldfinches are able to cling to the screen in order to feed. House Finches (Carpodacus mexicanus) will also visit goldfinch feeders.
Once you start regularly attracting Lesser Goldfinches (it may take a while), it's important to keep the feeder filled with thistle seed, or the Lesser Goldfinches may move on to a more reliable food source. Lesser Goldfinches also like to have a shallow water source nearby, and they especially like flowing or dripping water.