Appenzall

Easy and most impressive hike: Ebenalp
Take the quick train ride from Appenzell to the Wasserauen station, and change there to the cable-car to go up the mountain to Ebenalp. The trail takes you through a cave and around an ancient church in the cliff, originally for hermit-monks, then you get to a mountain restaurant/guesthouse built into the cliff. You can eat or drink, cliffside, on the outdoor cafe, overlooking the interesting rolling hills beyond. Then you can continue on the trail around the mountain and back up to the cable car station. Here's Rick Steves' description:
https://www.ricksteves.com/watch-read-listen/read/articles/high-in-switzerlands-appenzell

Here's the Ebenalp website with pictures and webcams:
http://www.ebenalp.ch/en/welcome

Here's the myswitzerland.com website with contact info for Ebenalp and other helpful references:
https://www.myswitzerland.com/en-us/ebenalp-through-the-cave-to-the-wildkirchli.html

I second the Aescher/Wildkirchli visit-- it was definitely a highlight of my trip to Switzerland. If you're up for a challenge, you may want to consider hiking from Wasserauen up to Ebenalp. It's challenging but beautiful-- up some steep hills and then through some rolling hills, then a rather flat section that passes by the Seealpsee (great place for a scenic picnic!), and finally a LOT of very steep stairs up to the Aescher Guesthouse and Wildkirchli. Then one more hill up to the Ebenalp cable car, which will take you back down the mountain. Gorgeous views through most of the trail and especially up at the top. I found it quite difficult, but somehow the difficulty made me appreciate the panoramic views at the top even more!