Here is a selection of excursions you can do from Brunnen:
A visit to Schwyz
Schwyz is very close to Brunnen. There is a lot of history – see tripadvisor.com/Travel-g188073-s2/Schwyz:Swi… -. From Brunnen I recommend warmly a visit to Schwyz. As you are there, you could also visit the factory shop of Victorinox, the owner of the make SWISS ARMY KNIFE.
There are many hiking paths from Muotathal available. You can contact the tourist office for suggestions - firstname.lastname@example.org -. The website of this tourist office - http://www.muotathal.ch/tourismus/ - is unfortunately only available in German.
Stoos is a resort above Schwyz. From the railway station you take the bus to the cable car. Up there you again have many hiking paths - see http://www.stoos.ch/sommerenglish.html -.
Treib / Seelisberg
Take the boat from Brunnen to Treib - seelisberg.com/index.php/gastronomie/91-wirt… - where the historic restaurant is and the funicular up to Seelisberg - seelisberg.com/index.php/ideen-fuer-ihren-au… -. Unfortunately these websites are only available in German. Use Google translation to translate.
From Seelisberg you can walk to Emmeten – Beckenried on different paths. From Beckenried you can take the boat back to Brunnen.
At Seelisberg you can find an Ayurveda health center.
At the end of the southern part of lake Lucerne (lake Uri) there is Flüelen where the Swiss Path passes. You can do a nice walk along the lake or have a pick nick or a bath there in the lake. Reach Flüelen by boat or railway from Brunnen.
There are many more excursions you can do from Brunnen