In Germany and probably also Austria, most trains stop running sometime early to late evening on both the 24th and the 31st of December. Service will not be resumed until Christmas Day or New Year’s Day morning. Typically, nobody will be outside on the evening of Christmas Eve unless it is on the way to church and back again. Plan accordingly. On timetables in Germany, trains that do not run on these days are typically marked with a small symbol which is explained in the legend as:
Nicht 24. und 31.12.
Depending on region and company, the morning and afternoon trains on the 24th and 31st will either run as on Saturdays or something between Saturday and weekday service. (In 2016, the 24th and 31st happen to be Saturdays, so there will only be Saturday service, anyway. If the 24th/31st happen to be Sunday, there will be a Sunday service with late connections reduced.)
On Christmas Day and New Year’s Day, trains run like on Sundays with a few not so important exceptions (e.g. certain trains that effectively only run pre-first day of the week and don’t run if Monday is a public holiday).
Local bus companies will follow pretty much the same rules; however, it can be tedious to find out which buses exactly run on those days.
For any leg that is in Germany and that you book via Deutsche Bahn, I would always recommend pre-booking as soon as possible as there is a limited number of special saver fares for as little as €19 on short legs. If you pre-book, you are bound to a train but you can also easily reserve seats (seat reservations are optional in Germany). The most comfortable option from far away would be their website bahn.de, in my opinion.
I am not sure how many people will travel on Christmas Day. In my experience, most people will travel home (or on holiday) on the days leading up to Christmas Eve which is when pre-booking and reserving should definitely be done. Thus, there shouldn’t be that many people travelling on Christmas Day. However, it is also the beginning of Christmas holidays after the main Christmas celebrations (Christmas is celebrated on the 24th in German-speaking countries), so the second wave on their way to winter holidays could be populating the trains.
As far as I am informed, there is no benefit to pre-booking Swiss and Austrian train tickets — but that information of mine could already be outdated.