The default texting app on the Droid X with Froyo (current operating system) does not show the smilies. You are not missing anything - Moto missed the boat!
You can install an aftermarket texting app called Handcent which will then show smilies. Handcent also has a lot of things in it that you can customize, so I prefer it over the stock app anyway. Just be sure to go into the Handcent setting and have the Notifications box checked so that the notification shows up in the status bar (I think this is checked by default anyway). Very important: also go into the stock messaging app and go to setting and make sure that the Notifications box in there is UNCHECKED, otherwise you will get 2 notifications for every text!
By the way, I recently installed the Gingerbread leak and this issue has been fixed - smilies show up in the stock app as well! (about time). I like the Handcent UI better, though, so I am still using it instead of the stock app..