Last summer I worked in Zürich, Switzerland. Of course, I took the train to get there (from Vienna to Zürich there are three direct connections each day, it takes about 9 hours) and of course I took my bike with me :-). Unfortunately, I couldn't use the night train, cause one is not allowed to bring a bike there. I also couldn't bring my bike on the first train in the morning, cause that was already booked out for bicycles more than one month before my travel. Luckily on the last train there was still some space available for my bike. The bicycle transport is quite cheap -- only 12€ for one way. On the way back I shipped it via rail cargo which was a bit more expensive -- 30 CHF (~20€).
In Austria, the new highspeed trains (the so-called ÖBB railjets) don't carry bikes at all. Actually that's against a guideline by the EU, but they don't care. Read more about it in an Austrian newspaper or on Radlobby.at (both in German). Moreover, in most other trains it's also either forbidden or complicated to transport bikes.
In Vienna, you're allowed to bring your bike in the subway at certain hours (not in the rush hour) for the cost of a half-fare ticket. Bicycles are completely forbidden in trams and buses though.
In Switzerland, public transport is great. I absolutely love it and I could talk about it for hours ;-). But I will focus on the bike-related stuff for the moment: bicycle transport in buses, trams and trains (mostly also on boats and funiculars) is allowed, cheap and easy. On the SBB website (Swiss national railway company) they also promote the transportation of bikes as hand-luggage in special bags ("Velotragetasche" in German), which in particular is free of charge and always possible.
When I lived in Switzerland, I thought about buying one of those bags, but in the end I didn't cause they are quite expensive (100-150 CHF) and because I'm probably not allowed to use it in Austria. If you're interested, you can download a review in the Velojournal here (in German, 2005).
But there are also other easy ways to transport bikes, e.g. the "Veloselbstverlad" in trains (for loading and unloading at the luggage vans no conductor is needed). Besides you can rent bikes for free in all major cities (but only for one day) and almost everywhere with Rent a Bike. Have a look at the map regarding self-service loading and bike rental.
Public transport and biking go quite well together in Switzerland and I think it deserves to be a role model.