Yes, there's a train station at the airport in Frankfurt.
The German train website is one of the better options for planning schedules for train trips in Europe:
http://www.bahn.de/international/view/en…
You can do this trip starting in either direction Frankfurt to Zurich or Frankfurt to Berlin. I wouldn't go from Munich to Zurich than back to Vienna and on to Frankfurt though. Frankfurt - Vienna - Munich - Zurich - Frankfurt or vice versa would be more reasonable.
There are discounted "savings" fares that are available if you buy the tickets far enough in advance. You can check specific dates for availability on the train website. For example, the standard fare from Berlin to Vienna is 116 euro on some trains, but there's a savings fare of 29 euro. Obviously, with this difference, the limited number of seats at the cheap price sell out quickly. There are usually some seats at other intermediate prices as well. You may not get the cheapest discount, but you still may be able to save money.