Amsterdam/ Rhine Area/ Munich/ Saltzburg

You might want to sign into Google Maps and enter your itinerary. I think you'll find the distances to be far greater than you know. I've made this route a couple of times by automobile--spending the night in Bacharach, 4 nights in Munich and 2 days in Salzburg. Vienna is another 4 day minimum stop as it's another very large and great European city.
You probably need to allow more time for such an itinerary--a lot more time than shown above.
We no longer drive as far and fast as we can on our European trips as we're moving slower and traveling "better."

You are underestimating the time it takes to do "quick cruises and train rides". Its over 250 miles from Munich to Vienna. Day trip? No, especially stopping in Salzburg on the way. Even if you flew it would be like a day trip from New York to Washington DC - no time to actually see anything.

there are issues with distances here - but in both scales.

Bacharach and quick cruises and train rides to the surrounding areas - what area? The Rhein between Koblenz and Mainz is a string of small towns and villages. The ship schedule makes it seem a long way but the local train calling at every station does the journey is less than 90 minutes.

You've got an interesting date to begin with. Saturday September 16 is "Rhein in Flames" night at St Goar . https://www.st-goar.de/819-1-.html almost every town on the wine growing stretch of the river does one of these at some point during the summer. St Goar will get busy, especially in the evening. The fact that you are heading for Bacharach suggests you don't know about this event! Ship schedules from late afternoon MAY be amended. I say may because the official KD Rheinschiffahrt website appears to have been hacked, none of my browsers or virus guards will let me near it!!

Anyway, given you're coming all this way it would be mad to miss the evening. So option 1 is try to get a room in St Goar not Bacharach, and do it now!

Train from Amsterdam is easy. ICE 105 departs from Centraal station at 08:05, arrives Köln Hbf 10:45. Local train departs 10:56 and arrives St Goar at 12:55. If you can stay in St Goar then do so, if not see Plan B below. The party actually begins at 11AM so will be going well by the time you arrive! Just make the most of the atmosphere and forget any more travel that day. Next day try the cruise and do a decent length to Bingen and then get a train back in you own time after a walk around town.

Plan B. If you can't get a room at the party try for Bingen http://www.bingen.de/en/tourism-culture/touristic-offers-and-services , it will be useful for onward travel to Munich. Same train from Amsterdam, wait longer in Köln for fast train IC2013 at 11:17, arrive Bingen 12:50. Check-in and drop bags. Then get a local train back to St Goar for the rest of the day. Last train back to Bingen will leave St Goar at 00:30 (half past midnight) Sunday morning. For a group of 3 to 5 people there is a special ticket on the local trains which will make the round trip cheaper (RheinlandPflalz Ticket). For 1 or 2 people ordinary tickets are cheaper.

Next day, see Bingen and then do the cruise up to St Goar. Local train back (one way tickets)

Without a pass book the train ticket Amsterdam to Hotel base as a through journey at bahn.de and do it soon.

September 18. What goes around, comes around - you need to be in Bingen! If you stayed in St Goard get the local train at 08:56. If you had to stay in Bingen you get longer in bed. Depart Bingen on EC115 at 09:52 direct to München Hbf arrive 14:11. As before if you need tickets book now at bahn.de and if from St Goar do it as a through journey.

Time for my lunch, I'll look at Salzburg/ Zurich options later.

Onwards!

Munich-Salzburg is simple with frequent trains. There are basically 3 choices, Railjet, Eurocity (alternate hours) or regional (hourly).

Railjet is non stop and fastest. Base price is €37 one way but discounts if you book in advance at bahn.de

Eurocity, a few stops including Munich Ost station which may be useful for hotel. €33 one way also discounts if you book in advance at bahn.de

The Meridian regional trains - several more stops but still manages to do it in the same time as Eurocity!! Base price is €31.10 but no-one really needs to pay that!! After 9AM (all day weekends) regional fares get slashed and without the need to book. If the main (Hauptbahnhof) or Ost stations are convenient for the hotel you just buy a Guten Tag Ticket from the Meridian ticket machine. €21 for one person - but you can add up to 4 more people to the same ticket for €6 each. https://www.meridian-bob-brb.de/de/meridian/tickets-meridian/guten-tag-ticket

If you need to use a tram, U-Bahn or S-Bahn (suburban) train to reach Hbf or Ost then you buy a Bayern Ticket from the machine at the stop. This is €25 and again you can add more people for €6 and it covers Meridian as well. It actually covers ALL regional trains and buses and city trasnport in Bavaria! Write names in blank boxes on ticket.

After this it gets complicated. Vienna and Zurich are both easy to reach but in opposite directions, make a choice!

Zurich is expensive but has the scenery all around (Luzern would be better), Vienna has history at the heart but is a bit dull around the edges.

Rome is way off course. Theoretically there is an overnight train from Salzburg but we have an ongoing debate about how unreliable it is https://community.ricksteves.com/travel-forum/transportation/stay-away-from-obb-trains

You could fly from Zurich, Munich, Vienna and sometimes Salzburg but it's a lot of hassle and timewasting with airports and check-in.

I encourage you to book your lodging in the St. Goar area ASAP.

Rhein in Flames is an amazing, must-see spectacle, with well over an hour of fireworks from castles on both sides of the Rhine, as well as from a barge on the Rhine itself. It was a highlight of our trip a few years ago. It would be ideal to spend the night in St. Goar, but if you can't, you can stay nearby (Bingen, Bacharach or even Boppard) and catch a late train back to your hotel that night.

Thank you for all your helpful information David, Stan, SteveB and Charlene.

I would add more time to your other stops and skip Rome. Seems out of the way from the rest of your itinerary, and it deserves its' own trip.