When we first moved to Tucson, we didn't see many goldfinches in our yard. But as soon as we put a fountain in the garden, goldfinches visited more often. Here's our homemade fountain that I put together years ago from an article in Sunset Magazine. At first we had the fountain in the front yard, but javelinas kept knocking it over at night. So now it safely sits on our patio in our wall-enclosed backyard.
When two finches are on the fountain at once, they seem to take turns getting a drink or bathing while the other one waits.
After getting wet in the fountain, the birds perch in the mesquite tree to dry off, and perhaps grab a snack.
Then, they visit the feeder. I recently hung this thistle sock feeder in the backyard and finches started using it right away. We enjoy hearing their "chee" sounds throughout the day.