A letter template must have a Data source of Trainings to be used for training reminders. In addition, for SMS messages the Preferred sending method must be SMS/text message, and for e-mails it must be Any mail or E-mail. If these conditions aren't met, VSys won't show the template as an option. Letter templates must also be configured on the Letters, E-mail and Mailing Labels panel before trying to send these.
Your reminder templates should probably include something about the training itself.
Note: VSys will strip out all formatting and images from SMS notifications as they're merged.