If your speech is designed to discuss a topic, beginning with an anecdote works extremely well. Write something such as: "It was a sunny afternoon. I was walking near the Eiffel Tower and wanted to get a drink. I went into a small cafe, but when I looked for my wallet to pay my bill, the realization hit. I was pickpocketed." This can introduce a speech on a trip to Paris or one on ways to avoid pickpocketing while traveling.
Use a statistic for an introduction to a speech that discusses an important issue. For example: "In 2006, 20,000 people died in the U.S. in alcohol-related traffic crashes." That should shock your audience and make it attentive to your speech on the dangers of drunken driving.
Writing an introduction with a startling statement grabs the audience's attention. You say: "I had three cataract surgeries." The audience wants to know how this is possible. This can lead to a discussion on how to eliminate medical malpractice and works for a convincing speech.
Ask the question, "Do you know where your children are?" You can get your listeners wondering if they are taking good care of their children. The speaker can cover the topic of parental responsibility and control. Use the question to discuss an important issue for an audience of parents or educators.
Are you writing an introduction about sports? Or perhaps you are writing one on the successes and failures of life. In either case, write the introduction using the quotation, "But there is no joy in Mudville--mighty Casey has struck out."
If your purpose is to entertain, you may want to write a speech starting with a joke. This will relax the audience and pique its interest.
The speaker should bear in mind you would not use the same introduction to address a Girl Scouts convention as you would to a group of senior citizens. Likewise, an introduction to a graduation speech would not be the same as one for a eulogy. The introduction must be appropriate for the occasion.