Making a habitat for birds in my yard is one of the most interesting and satisfying things I have done. During mosquito season (April-November), I do not frequently maintain a bird bath, but the birds love it when I do. Put a rock in the middle so they have something to stand on. Here are some items I prefer.
During migration season in Austin, Texas, mid-April to mid-May, I use
this link to track nighly movements. Yes, they fly at night to avoid predators. The website uses satellite data to predict the density of birds flying over.
Mosquitoes
To be clear, leaving any standing water e.g., a bird bath, a saucer under pot (which I don't use), or water on any object including agave leaves, will accelarate mosquito growth. I have a massive invasive bamboo forest at the west border of my property which I know promotes mosquito growth since water and dew on the bamboo leaves or collect in the downed leaves.
I may try a monthly service to reduce mosquitos but for now use
a Thermacell Patio Shield (which has a 15 foot radius and uses a self-charger and replaceable liquid cartridges) or
Skeeter Screen incense sticks which smell nice away and last many hours. Note the Thermacell is not effective when there is a breeze but having said that, mosquitoes are bad flyers and can't fly in a breeze.