Rail Tickets vs. Rental Car

Travel by train would be easier and a more efficient use of your travel time, especially when using the high speed trains which travel at up to 300 km/h. You can certainly buy your tickets a day or two before you'll be travelling. Note than many cities in Europe have more than one station, so you'll need to know which stations you'll be using when you buy tickets. You may have already seen this but if not you may find it helpful - https://www.youtube.com/watch?v=3AxtbifeoBY .

A car will be more of a hassle, especially at that time of year as you could encounter winter conditions. A few more points to note....

• Renting a car in one country and dropping off in another often comes with a hefty surcharge.
• Each driver listed on the rental form will require an International Driver's Permit for driving in Austria. These must be used with your home D.L. and are valid for one year.
• With a car rented in Amsterdam, you'll have to purchase an Austrian Tax Vignette as soon as you cross the border, or hefty fines may result. These are collected on the spot!
• As you're travelling only to larger cities, parking is likely going to be expensive so trains would be a better option.

Yes, you probably do save money buying train tickets in advance for German trains. Cheaper fares tend to sell out closer to departure. Depends on which train, how busy it is that day, etc. But buying the cheap fares means you have limited flexibility to change your plans.

You could buy a German rail pass, which would allow you to choose any train at random on days you want to travel. That's probably not cheaper than buying your tickets weeks in advance, but it's probably cheaper than buying the last-minute tickets. (I think German trains can't technically sell out, but if you want a reserved seat, those may sell out; without a reserved seat, you may be stuck sitting in the dining car.) You have to buy the pass by mail or only at certain German train stations.

Here's some more info on German trains and passes:

https://www.seat61.com/Germany-trains.htm#Railpasses_for_Germany
https://www.seat61.com/international-trains/trains-from-Munich.htm#Munich-Amsterdam
www.bahn.com

Vs. a rental car? I've driven in Europe a few times, but I greatly prefer the trains, and Germany has an excellent rail network (far superior to anything in, say, North America). I'm also not sure I'd want to drive in November-December when days are short and the possibility exists for bad weather. If you rent a car, check prices - could be that renting in Amsterdam and returning in Germany incurs an extra fee (vs. renting in one German city and returning in another). I'd take the trains, personally.

Tickets bought nonrefundable now for about half the price of the walk up price. Amsterdam to Cologne for about 70 EUR vs 140 EUR for 2 at the last minute. Cologne to Nuremberg for about 100 EUR vs 180 EUR last minute. Nuremberg to Salzburg 40 EUR vs 146 EUR. Salzburg to Munich can be done with a flexible Bayern Ticket for 31 EUR for 2 in less than 2 hours, not much slower than with express trains.

Tickets and schedules at www.bahn.com.

Amsterdam to Cologne, I'd also spring for 4.50 EUR per person seat reservations. That route can be busy. Check the price of 1st class, that includes seat reservations. When you buy advance tickets they are often just a little more than 2nd class.

This should be a no-brainer. Train is much better than a car for this trip - a car would just be an expensive headache. As long as you are confident in your plan and schedule, buy non-refundable tickets far in advance and save a bundle. I like to drive but I'd buy train tickets for this route.

My answer to your question is "and" instead of "or".

"Do we save money by booking tickets in advance?"
Yes, look at fare options on website of Deutsche Bahn. Sometimes you save more by taking the bus, e.g. FlixBus.

Biggest issue with rental car from Amsterdam is that it has no emission class sticker from Germany that you need to enter German emission zones in listed cities (hot topic currently), e.g. Cologne. You can buy it but do you want to care for it?

Rule of thumb: For city hopping in Germany trains and buses are the easier option, likely most times also the cheaper for 2 persons only. Public transport in German cities is excellent. If you want to explore countryside a rental car makes sense, e.g. stopping at Adidas /Reebok or Puma outlets in Herzogenaurach.

My suggestion: Take trains / buses and consider rental car between Cologne and Nuremberg ;-)

The 7:53 ICE from Cologne Hbf direct to Nuremberg is about an hour slower (4 1/2 hours) than the fast route, but it goes along the Rhine to Frankfurt, so you get to pass through the middle Rhine valley, with all the castles and the Lorelei. The fastest way is to cross the river on the S-Bahn to Cologne Messe/Deutz and then on an ICE that uses the high speed tracks to Frankfurt. Definitely less scenic and tickets are more expensive.

A 3-day German Rail twin pass (flexi version) will cover Cologne > Nuremberg > Salzburg > Munich for €270 for two adults under the current Autumn Special promotion at DB. Lots of flexibility with the GRP. Use any trains you like at any hour on those 3 travel days of your choice. You can choose your days as you go if you please; if your plans change now or later, take journeys to different places on different days as needed. Take additional journeys free of charge on those 3 days if you wish (you could for example stop in Berchtesgaden on the route to Salzburg, or after the 1.75-hour trip to Munich, take an outing to Augsburg or Landshut.)

Alternatively, pay €225 for a two-day GRP flexi (Cologne - Nuremberg - Salzburg) and buy a "Bayern Ticket" day pass for two for Salzburg > Munich (€31 at the ticket machine, travel at any hour on regional trains, except only after 9 am weekdays.) That would be cheaper.

Then look for an additional saver fare ticket at DB for Amsterdam > Cologne.

Always good to look at the resources of the RS Site, where he discusses "Rent a car or take the train?":

https://www.ricksteves.com/travel-tips/transportation/car-or-train