Begin storytelling as early as possible. Make storytelling part of your infant's day. Pick a quiet time when baby or toddler has wound down from his activities to read to him. Snuggle up with the baby in a comfortable chair and begin telling him all about the "Poky Little Puppy" and "The Truck that Drove All Night". Don't worry about his level of comprehension at this point, instead trust that the rhythm of the language and your warm and expressive voice engage him.
Choose age appropriate materials for your children. Pick the very best literature possible. For the very little ones, try a title such as "The Very Hungry Caterpillar". Preschoolers will enjoy more raucous material like "Chica Chica Boom Boom". "Mr. Popper's Penguins" is best enjoyed by the early elementary crowd and the extremely well written "Holes" works exceptionally well for the later grade school years. Tweens and teens love "Harry Potter" of course and many of them are ready for award winning serious literature like "Chandra's Secrets" and "The Giver". A resource for finding great stories is the 100 best books listings at the teachersfirst website which includes age recommendations.
Preview your material before presenting it to your audience. It is very helpful to know the background of your story to make for a smooth telling event. Whether you are reading just to your own children or a larger group, it will give you more confidence to have the background information in place. You may even wish to "learn" stories for telling in which you do not use the books at all but rather present simply with your voice and gestures.
Act the story out with your voice and facial expressions as you read to your children. Get inhibited as you create different voices for the characters. Be sure to capture the emotions of the story with your tone. Read exciting scenes faster with a heightened voice while mellowing for restful interludes of the story. Make your kids laugh with silly stories like the "Lemony Snicket" series by being silly yourself and let them emote with you when you read sad passages from tearjerker like "Bridge to Terabithia".
Make up your own stories if you are so inclined. Do not worry about the literary merit of your tales especially when storytelling to very young audiences. Experiment with material from your imagination, perhaps even taking beloved characters such as "Dora the Explorer" and "Spongebob Squarepants" and devising your own adventures with them. The children will love it.
Speak clearly and loudly while storytelling. Remember as you are entertaining you are also teaching and modeling good oratory skills will help children be more successful with this skill. Also, you will quickly lose attention if your voice is too soft or your words muddled.
Tell beloved stories over and over. Children love to hear their favorites and with the practice you will become adept at your particular tales. Involve your children with these "repeats", letting them join in with familiar refrains such as "an elephant's faithful 100 percent" from Dr. Seuss' "Horton Hatches an Egg".