Next month as in March or as in April?
While weather can be nice in either, it can still be very cold too. End of April can be perfect hiking weather, or still having some snow on the hills.
Lindau/Bregenz is probably more of a summer destination than winter, but you'll find some nice things to do for sure. Most of my suggestions will be on the Swiss side, though (will you have unlimited rail pass? then it's not really an issue..) - public transport in Switzerland is much better than in Germany/Austria, as you'll find out.
On a really bad day (rain/cold), you might want to go for a hot indoor fun bath - Säntispark. Basically a "six flags" in the water. Newly renovated and expanded. If you're into that.
Alternatively, the "cheese production museum" in Stein AR (Appenzell AR) might interest you, since you mentioned Cheese.
http://www.schaukaeserei.ch/en
Or the pitoresk city of Appenzell itself. Also, St. Gallen has a nice old town (it can all be combined into a long 1-day trip) too, and many interesting historical buildings.
For hiking, it will HEAVILY depend on the weather (and the weather in the weeks before) - most probably, it won't be really nice to go on a hike in the hills around Bregenz so much yet. You might want to check the weather shortly before departing and thus plan on the go.
I'll most probably be in the area too, so if you like, I can guide you a bit around myself. Hit me up if interested..