Landing in Vilshofen-then what?

How difficult is it to get a train from Vilshofen to Munich?

No problem. There is a direct regional train from Vilshofen to Munich hourly. It is the fastest connection at just over 2 hours. It originates in Passau about a quarter hour earlier.

The two of you can do it for 32€ with the Bayern-Ticket, but there is also the Regio-Ticket Donau-Isar for 2€ less. It's just like the Bayern-Ticket except valid for a smaller geographic area.

have another 7 open days to travel “Romantic Road”. ... Husband wants to do train

Although there are trains connection between some of the towns on the Romantic Road, much of the Road is only connected by buses. There is the Romantic Road Coach, but it does the entire Road in one day and doesn't stop long enough in the towns to really see them. There are also local buses connecting all of the towns on the Road. In particular, the stretches of the Road from Füssen to Landsberg am Lech and from Nördlingen to Weikersheim are accessible only by bus.

As a road trip, the Romantic Road is nothing special. It's just an ordinary country road - two lane and clogged with truck and farm equipment. It's not about driving the road; it's seeing the towns along the road. Some are more interesting than others. I particularly enjoyed Landsberg, Nördlingen, and Weikersheim (I only stopped in Rothenburg long enough to have lunch, as I had already been there twice).

Some people will tell you you have to have a car to do the Romantic Road, but that is not true. I did the Road in 2007 using public transportation exclusively. I started at Wieskirche (I had already been to Füssen several times) and did the whole Road, including the loop from Rothenburg to Würzburg via Weikersheim and Bad Mergentheim. I spent nights in Landsberg, Nördlingen, Feuchtwangen, and Weikersheim. From end to end took about 4 calendar days (3 full and 2 partial), but I spent only a few hours each day (Ave) traveling between towns and most of my time seeing the twelve of the towns on the Road. Prior to my trip I did a lot of research to figure out which town I wanted to see and the transportation schedule between them. If you don't want to take the time doing the research (or you don't know how), then renting a car might be your better option.

there is also the Regio-Ticket Donau-Isar for 2€ less. It's just like the Bayern-Ticket except valid for a smaller geographic area

... but it is NOT valid for urban transport in Munich, which makes it a rather bad deal.

Some people will tell you you have to have a car to do the Romantic Road, but that is not true.

Well, it's not true but ... having a car allows you to make more of your trip at least along the Donauwörth - Würzburg segment:

Dinkelsbühl and Öttingen (the architecture of which reflects the confessional division of it) are difficult to reach by public transport from the south (no problem from the east - Ansbach), almost inaccessible by public transport are Neresheim Abbey, Schloss Schillingsfürst, Creglingen (the Herrgottskirche hosts the absolute masterpiece of Tilmann Riemenschneider), and Stuppach (the village church hosts the famous Stuppach Madonna painting by Matthias Grünewald, well known for his Isenheim altar in Colmar). Personally, I'd rent a car in Augsburg and drop it in Würzburg. Between Nördlingen and Würzburg the route cuts through the sparsely populated Western Franconia, so parking would not be a problem anywhere.

Thanks, these responses are helpful. Another thing we are pondering, should we make lodging reservations prior to our trip, or can we “wing it” as we travel from Fussen to Rothenburg up to Bacharach?

At that time of the year, long before the Easter holidays, there is certainly not much going on. I would book accommodation for Füssen upon your arrival in Munich. For the rest I'd book always one day in advance via booking.com or HRS.com.

(The cheapest accommodation can usually be found on the websites of the local tourist informations. However, they offer rarely electronic booking options)