Hotel Schoenburg

Go to the German Rail site at http://www.bahn.com/i/view/USA/en/prices/index.shtml
Specifically "Saver fares" under "City to city Tickets" and "Tickets for Regional Trains". Buying long distance train tickets can be cheap if you get nonrefundable tickets in advance. They are priced like airline tickets. Buy early and save big, but you are committed, as there are change fees.
Regional and national day tickets are available for groups of up to 5 traveling together giving unlimited travel for an entire day after 9 am or all day on weekends. These tickets include local transport. There are also local passes within cities or regions covering smaller areas that include local transport.
The Burghotel Auf Schoenburg is not far from the train station, but quite a ways uphill. There are taxis. I don't know if the hotel provides shuttle service, but it wouldn't hurt to ask.

I can't comment about that specific hotel, but usually if a hotel sits at a difficult-to-reach location (like on top of a hill), they will usually operate a shuttle.

RE- Rothenburg odT... despite what the tourist literature claims, this town isn't nearly unique as advertised, except for the fleet of tour buses that visit daily and all the trinket shops. Visit if you really must, but there's other options more convenient to your trip. For example, if you want a preserved walled town, the beautiful twin wine/spa town of Bad Neuenahr-Ahrweiler is just off the Rhine near Ramagen.

We stayed at the hostel part of the castle. Beautiful setting! We had a peek at the hotel, it's very impressive. Do have dinner on their patio, the views are incredible!

It is quite a steep way up! Nice for walks down to town, especially if you take the path through the vineyards. However, you don't want to carry luggage uphill. Ask the hotel if they provide a shuttle service.

We stayed at this hotel two years ago, and found it to be a most memorable (albeit pricey) experience. You can select your specific room off of their website. We opted for the room that has a "bookcase" door to the bathroom. It was the size of a small apartment, with lovely views in every direction. The included dinner was a delight. You receive a small bound book listing the meal for that night. Many courses with options offered if you didn't care for one of the choices. The hotel did not run a shuttle, but when we arrived at the train station we called for a taxi. I think there may be only two taxi drivers- a man and his wife. Very friendly and helpful. We walked down everyday, but always took the taxi back up the hill. Once you get to the hotel parking lot you can request to have a man from the hotel come and get you and your luggage for the last big uphill trip to the hotel itself. It sounds complicated but really works efficiently.
We had a rail pass, and it worked well for us. Our pass included boats so we could make day trips to other small towns like Bacharach. We also listened to a RS podcast during a river ride and it was like having our own personal guide. Check the DB website for specifics about what modes of transportation your pass includes. The Oberwesel station is very small and not as frequently served. When we were there local flooding had delayed/cancelled some trains and if we had point-to-point tickets we would have missed our connection to Paris. Since we had a rail pass we could just hop on the next available train and still get to a connecting point. Good luck.

Thank you SO much for all the information. And thank you Tom for your insight regarding Rothenburg. We will check out your suggestions. I suspect we may want a break from touristy 'stuff' by the time we are heading toward Rothenburg.