Absolutely do the whole thing by rail - rail is great in Europe and very cheap with the eurail tickets. Driving, on the other hand, is extremely stressful if you do not know where you are going.
Completely agree too with the other people that you are trying to cram in too many places. What is your theme? Are you after culture? Or scenery? Or food? Or "just said I've been there"? Different places are good for different things. The best of switzerland, for example, is NOT in the big cities at all - but in the smaller towns/villages/mountains. Zermatt, for example is beautiful and not too far off the high speed rail routes. (the train to zermatt is an experience in itself). However, nobody goes to Switzerland for the food ... (sorry all the swiss on here, but compared to france, italy etc, swiss food is horrible!).
I could (and have) spend 2 weeks by rail in Switzerland alone, so just a day here and a day there ... you'll be SO tired and nothing will register.