There are two approaches to imitating bird whistles. One is to whistle and use other sounds to imitate birds naturally, which can take years to master. The other is to use a bird whistle, a small musical instrument that can imitate many different bird sounds with practice. Some bird whistles are impossible for humans to mimic naturally because we do not have the mechanisms.
Listen to the bird whistles you wish to imitate. ENature.com has dozens of different bird calls (see Resources). If it's a songbird and sounds similar to a human whistle, you can probably imitate it naturally. This is easier for women or children because of their smaller whistling apparatus and ability to whistle higher notes.
Copy the bird whistle note by note by whistling along. Even if you don't think you have a good ear for pitch you should still be able to match pitch with a few tries, as it is the same listening action that allows you to tell different people's voices apart. If it is too high, try whistling it lower with the notes the same distance apart.
Purchase a bird whistle for more complicated calls or sounds that humans cannot reproduce (see Resources). It will come with instructions and suggestions. Place the centre of the bird whistle up to your mouth, cover the opening completely with your lips, and blow a gentle and steady stream of air. Experiment with faster and slower airflow, alternating puffs of air and changing the angle of the whistle against your lips.