Mosel Favorites?

"We're hoping to find somewhere with easy train and/or bus connections for day trips, ideally including morning rides up or down river followed by hikes back."

Sounds like a nice strategy. I assume you want to stay ON the river. Almost all riverside towns on the upstream half of the Mosel River (between Trier and Traben Trarbach roughly) have only bus service. Since you have 2 weeks, I'd definitely want a town that has train service (not all do - it's a very playful, windy river and the tracks take many shortcuts, bypassing a lot of towns.) This means staying between Traben-Trarbach and Koblenz. There are two train lines along the Mosel - the main Koblenz-Trier route, and a short trunk line that leaves that route in Bullay for Traben-Trarbach. Have a look at THIS MAP - all the main line stops from Bullay to Koblenz, and the trunk line stops from Bullay to Traben-Trarbach, are right on the river. The map also shows how the Mosel is connected to the Rhine River towns (Bonn to Bacharach) and the Lahn River towns (Koblenz to Limburg) via the area's train hub, Koblenz.

If you pick Bullay, Alf, or Reil (all 3 are small, quiet, non-touristy towns with train station) you will have some lovely walks right in your midst. Go to THIS PAGE for a map of the Moselsteig segment there (in German, scroll down to map.) THIS PAGE in German has description and photos of the Zell-Neef segment (which includes Bullay) and an English translation option. The Bullay/Alf/Reil area where the river nearly doubles back on itself is wonderfully scenic. You can hike to Burg Arras castle (castle-hotel with small museum, refreshments) outside of Alf or to the Marienburg and the nearby Prinzenkopf lookout for a commanding view. Bike paths take you around the big curve to Zell, Pünderich and beyond. Note that Alf has no station but is immediately across the river from Bullay (ferry crossing) and if you stayed there you could use the Bullay station. For rooms/apartments check the ACCOMMODATIONS search page from the LOCAL TI.

Some folks will may suggest you stay in Beilstein, which is very small and cute and has become very popular with the Rick Steves crowd. Note that it's bus or boat only from there, however. It's a town you might want to visit on an outing, but I think stuff to do is more abundant, and the getting around easier, if you stay elsewhere.

Other lovely towns to consider: Cochem and adorable Winningen (the latter if you want a better location for exploring the Rhine too.)

Here is the site with maps for biking in the Rheinland-Pfalz state in Germany, which includes the Mosel.
They have a bus service which carries a bike trailer you can use with a paid reservation. This shows the bus routes and some of the trails and even what the elevation changes are along the trail.
http://www.regioradler.de/index.php?pre=en_&haktiv=10&code=

Here is the Moseltal Tourist website with event calender and accomodation booking engine:
http://www.mosellandtouristik.de/en/wine-cuisine/wine-events/wine-festivals/

I strongly advise you to stay in a town that has a "Strassenweinfest" over the weekend. These events are a blast with town bands playing and all the local wineries have booths selling wine and federweiser, which is unfermented or partially fermented grape juice. If some bands from Holland show up, your in for a treat because they are "wild and crazy guys (and gals)". Mosel Valley is a popular weekend destination with the Dutch. My crummy German led all the locals to believe I was Dutch at first. Not too many American tourists here.

For accomodation, look to stay in a guest house, especially a Weingut. We stayed in one in the town of Briedel. Lower floor/basement is the winery with tractor storage, grape press, fermentation tanks, and bottling equipment. First floor is Kitchen, 2 guest rooms with en suit shower and such, and living room/breakfast room, and large outdoor veranda. Top floor is family living area.

Briedel is not the most convenient location, but we chose it because the neighboring town of Puenderich had a Strassenweinfest that weekend. Had to get a bus from Bullay on the main Trier-Koblenz rail line.
You might look at renting bikes for your time there. Although our Weingut had 2 bikes to rent for the day, we had to share with guests in the other room. Great way to get around as there is a nice bike trail all along the river in an old railroad bed. You can also take bikes on the train. Closest train station is Reil. Only problem is it is in the Trier Verkehrsverbund, and Briedel, Zell, and Bullay are in the Rhein-Mosel Verkehrsverbund. The VRM Minigroup ticket gives you all the trains and buses from Bullay to Koblenz and up and down the Rhine from almost Bonn to Oberwesel. I got busted for riding the train to Reil. Fortunately just had to pay the conductor the extra fare.

The Mosel valley was the lowest cost and most fun time of our German holiday.

"They have a bus service which carries a bike trailer you can use with a paid reservation."

Sam's comment reminded me of the fact that in the VRM zone (between Bullay and Koblenz on the Mosel) you can take your bike on the local trains (or buses) for free (unless it's before 9 am on a weekday - then your bike can go on the train only and needs a ticket too.) From the VRM pages:

"Cycle ticket: When you travel on the VRM network you can also take your bike along. After 9 a.m. Monday to Friday and on Saturdays, Sundays and public holidays, this service is absolutely free!"

I've based in Treis-Karden, two towns across the river from each other, with connecting bridge and good train connections. Hotel Brauer worked well for me. Bike rental place was two bocks away. Easy bike rides to Cochem and Burg Eltz (bike and then hike to the castle) and all the other towns along the Mosel. Stayed only one night on my last visit (on way to Trier) and had excellent dinner at Hotel-Weinhaus am Stiftstor.
I love the Mosel area!

Thanks so much for the responses and the great information. I'll have to parse through the links tonight. We've never been to the area and are really looking forward to it.