In prior versions of iOS, users were able to adjust the font size for specific core apps through the Accessibility menu. Those apps were Mail, Contacts, Calendar, Messages, and Notes. Sure, this was helpful to those who had trouble seeing the default text size on an iOS device, but outside of those apps it was of no help.
To change the size of Dynamic Text (its technical name), launch the Settings app and then select "General" followed by "Text Size."
You'll see a slider with some sample text just above it. Reposition the dial to increase or decrease the font size, using the sample text as an example of the font size you've adjusted to. When you're happy with the size, exit the Settings app.
Going forward, any apps -- including third-party apps -- with support for Dynamic Text will then display your text using your preferred font size.
As we begin to see the influx of apps that are properly updated to iOS 7's APIs, we will also begin to see apps that support Dynamic Text. If you have a favorite app you would like to see support this new feature, be sure to reach out to its developer and let them know. In the end it will not only benefit the users of the app, but the developer as well.