The ride up counted as sight-seeing for quite a bit of the ride. I was playing Professor Layton, and I kept having to set the game aside to look at stuff and take pictures. We kept saying to one another, "Do these people realize how good they've got it?"
We didn't have much in the way of elaborate goals for exploring the city. Going by the gestalt of the stuff I'd read online, we decided to just wander down through the old city to the big churches, then come back up on the other side of the river and see the National Museum.
Zurich is pretty cool, but I like Bern better. Zurich is definitely bigger and more happening, and it felt more car-oriented than Bern. The old city is pretty neat. It's very tall and very close, and I imagine you could spend hours trying to find that one shop you saw the last time you were there.
One rule of exploration we have discovered on this trip: if there's a narrow street or passageway that goes up a hill, you should follow it, because it probably goes somewhere interesting. In this case we found ourselves up on top of the Lindenhof, which has some really great views. It also had dozens of people playing petanque (the French version of bocce).
Anyway, we made our way down to Fraumunster, where there are these famous stained-glass windows by Chagall. They still had services going on, so we crossed the bridge and had a look at Grossmunster instead. (Didn't really feel the urge to go inside; it's plenty interesting from outside and we were a little churched-out at that point.) We encountered an amazing busker playing Vivaldi on an accordion, so that was pretty entertaining. Poked around the shops, found a Starbucks where we could use the toilets and sit down to rest our feet for a bit.
After the break, we went back to Fraumunster and saw the windows. They were pretty, but honestly I like traditional stained glass better. Still worth seeing, though. We then started hunting for lunch, doing the "nice-restaurant lunch special" thing, and found a place serving schweisschnitzel. So we have now officially had the native cuisine! It was tasty, although I am starting to crave some kind of stir-fry with lots of minimally-cooked vegetables in it.
The National Museum is pretty cool. It's a museum of All Things Swiss. It's in a big old castle-y building (though they're hard at work on an expansion). There was lots of stuff about history that we just sort of skimmed through since we couldn't read any of it. As with all European museums, there's a LOT more stuff made of gold and silver than you find in American museums. The layout aesthetic was funky and nontraditional and I bet my Mom would be really interested. We spent the most time and attention in the Tower of Arms, or Waffenturm! Halberds and swords, is what I'm saying. Aw, yeah!
And with that, we called it a day and came back to Bern.