I am in Zurich on business in late May. I have two free days (in a row) to travel. I want to rent a car and drive into the mountains. I would appreciate any tips or your thoughts on "shouldn't miss" opportunities.
You shouldn't miss jumping on the train in Zurich and heading either to Lucerne or perhaps even farther to Lauterbrunnen. Why do you want to rent a car?
Provided the weather is nice, I'd go to the Lauterbrunnen area in the Berner Oberland. But I would do it by train. You don't really need a car because the public transportation is so thorough. And once in Lauterbrunnen you would park your car and use cable cars and funiculars to move up and down the mountains. Remember, the Swiss put the fun in funicular!
The SAC-CAS hut system.
There is a lot to see in Switzerland from small towns (Gruyere, Muerren, Gimmelwald, Grindelwald, Engelberg--and there is a variety of small to large even within this category) to small cities (Luzern, Zermatt--yes, Zermatt), areas/regions (Appenzell, Bernese Oberland, etc.). In these areas there are BE-UUUU-TIFULLLLL things. Funiculars to take you up and down into the mountains. Cogwheel trains to get you to the funiculars. Normal trains (but cute and Swiss) and boats (small ships) to get you to the cogwheel trains. EVERYWHERE.
I love the cheese and have a little lady I stop by and get a wheel to eat on the road.
I love the way the Swiss raise and care for bees--Bienenwagonen and Bienenhausen. Bee wagons (with huge wagon wheels and, like 20 hives inside) and bee houses with 20-50 houses. It really is unique in the world. I have done some research in this area... ;)
I love the wine and locally produced liquors.
But, after all of that, I love the SAC-CAS hut (huette) system.
http://www.sac-cas.ch/en/huts.html And click 'Search a hut' in the right hand panel.
The huts (152) are owned by the individual sektionen of the SAC, Swiss Alpine Club. Some sektionen own 0, some own 1 or 2, and some own up to 7. Each is a different hut with an individual design, sometimes required by the location :) Check them out. You may not want to visit the Monte Rosa (http://www.section-monte-rosa.ch/de/unsere-hutten/monte-rosa-huette , scroll down for pics ) but it is a great example. Some are very traditional some are bleeding edge modern (we stayed in Basecamp Matterhorn, a temporary 'hut', http://www.hoernlihuette.ch/base_camp.html ). That was definitely a 'don't miss'!
Enjoy!