I’m embarrassed to admit this, but for over year after switching to the iPhone I missed those silly little icons from my older phones. But it turns out that the iPhone had them, but they have to be enabled.

Here are the instructions:
Go to Settings
Go to General
Go to Keyboard
click on International Keyboards
click on Add New Keyboard…
Then choose Emoji

The phone will walk you how to switch keyboards when you are sending text messages.