Sorry you didn't get any info earlier - I haven't covered exactly the same territory, but it looks like there's a train from Bari north through Foggia and on to Naples, and from there to Sorrento. There's also a more complicated route Bari-Taranto-Metaponto-Battipaglia-Naples but it looks like a lot of changes. The Bari-Foggia-Naples route seems to be sans transfers. I'm looking at a very old and dog-eared Thomas Cook European timetable, an updated copy of which you can order on line or possibly through a local Thomas Cook travel agent - but for a short stay it's probably not worth your while.
A good resource for checking current train routes and schedules is the German rail website www.diebahn.de
, which does have an English option if your German isn't great. Just type in the start and end points of your journey and it'll give you all the route options you could wish for.
Don't know much about bike tours in France - do you specifically want to join a group, or would you be comfortable going on your own? I would imagine just renting a bike wouldn't be outrageous, but haven't done any bike touring myself. I assume you've taken a look at any recommendations in "Let's Go" or "Lonely Planet" travel guides - they seem to be the best for budget recommendations. Frommers and Fodor's are a bit higher end.
My experience with Eurail passes has been good, in terms of value and convenience, but it does depend on what you plan to do. I don't believe the Czech Rep is included on the western European passes, but it's relatively cheap and quick to get to Prague from Vienna, so it's still worth going.
Is your 22 days including the week in Italy and your time in Greece? If so, you're probably only looking at about 10 days in the other countries you mentioned, which might be a bit of a rush. 4 countries in 10 days, with travel time to consider too, you're looking at about 2 days per country. If you want to stick with those countries, I'd suggest just visiting one city for a couple of days, otherwise you'll be spending your entire life on trains and at stations. Three days per stop is usually a good minimum for seeing stuff and getting a feel for a place, so if you could either cut down on stops or extend your time, you might find the pace a little less frantic.
Once you've got an idea of your itinerary (circle routes are good if you've got to fly in and out of the same city, and zigzaggy routes are generally the least efficient in terms of travel time), check how long the trips will be (at the die bahn site). If there are any longish ones, consider an overnight train to save on both accommodations costs and sightseeing time), work out how many actual days of train travel you plan on doing, and then check the Eurail options. I know on the East Rail pass you have the option of buying extra days, if the passes offered don't quite meet your needs, don't know if that's the same on the western passes. If you plan on visiting two places that are quite close together, it might work out cheaper to just pay that ticket point to point and save the pass for a longer journey. Travel guides often give a good idea of fares between specific destinations (Naples-Sorrento, for instance).
Good luck with the planning and the trip, and I hope this info isn't too little too late.