Hi Annie,
I found the fares on kayak.com, departing May 24 and returning May 31. (I just chose two dates near the end of May - I haven't checked how available these fares are on other dates.)
For Chicago to Zurich, it shows several itineraries on airberlin at $849. That's the price through various ticket aggregators - the price through airberlin itself is $860.
For Denver to Zurich, it shows a bunch of "hacker fare" itineraries priced around $1050. (You may need to scroll down and uncheck the "hide longer flights" box to see them all.) Basically, get yourself to Chicago by whatever means you can, then take airberlin to Zurich. You'd need to leave yourself a ton of time in Chicago, though, because when you're on two separate tickets you have no protection if your flight is delayed and you miss your connection, which is why so many of the flights are so long.
There are also some hacker fares that make use of an SAS flight from Chicago to Zurich via Stockholm (outbound) and Copenhagen (on the return). They're about $100 more.