Evernote (freemium) is the Big Daddy of note-taking apps, despite increasing competition from Microsoft OneNote. Using 3D Touch on the app icon, you can quickly create new notes, take photos, set reminders, or search for existing notes. As of this writing, Microsoft OneNote doesn't support 3D Touch.
Handle (free) is a useful app that merges to-dos, Gmail, and calendar appointments. With a hard press on the app icon, you can create new to-dos, email, or events. You can also preview email. Handle is a universal iPhone and iPad app, and it has an Apple Watch extension, but it's currently limited to Gmail.
Spark (free) is a quality email app for iOS, though it's not optimized for iPad as of this writing. With 3D Touch, you can view your calendar, a list of recent email attachments, perform searches, and start a new email by hard pressing the app icon. Within the app, 3D Touch lets you preview email, as well as choose to reply, forward, or archive or mark messages as read.
The Weather Channel (free) just received a major upgrade, with 15-day forecasts, flu and pollen reports, and the ability to remove in-app ads for $4 a year— yay! The app also got 3D Touch support. Force-touching the app icon lets you jump right to the app's radar feature, search locations, or check the weather for two different locations.
Citymapper (free), a GPS app that saved my sanity last year in London, makes particularly good use of 3D Touch. Pressing down on the app icon lets you quickly view nearby transit stops and begin navigation. Within the app, 3D Touch lets you view walking maps to transit stops.
OpenTable (free) uses 3D Touch to give you fast access to restaurants you tagged as favorites, and it lets you view upcoming reservations. Within OpenTable, 3D Touching restaurant names displays their locations on a map, along with upcoming available reservation times.