How do I get birds to fly around my Island?

You have to complete all the landmarks (Lighthouse, Chapel, Pirate Ship, Mountain Chateau, and Aviary). After you complete the Aviary, you will be able to select bird eggs, hatch and release them on your islands. For estimated timeline, see schedule below:

If you renovate every 24 hours:

Lighthouse = 141 days = 4.7 months

Chapel = 171 days = 5.7 months

Pirate Ship = 190 days = 6.3 months

Mountain Chateau = 228 days = 7.6 months

Aviary = 60 days = 2.0 months

It's faster if you renovate every 20 hours.