A good resource to search out trails is the Rails to Trails Conservancy's "Traillink" website: http://www.traillink.com/
We've done day biking and even overnight tours with our son since he's been a bit over a year old (he's nine now). Lots of fun and a great way to explore. Hiking is good, too, of course, but you'll see a lot more on the bike. You just have to decide whether you will use the bikes enough to warrant bringing them. We did a cross-country tour with our campervan last summer and debated whether to bring the bikes. In the end we did, and I'm glad. We did rides in every national park we went to, and lots of other places, too. In a way it forced us to ride more, because had this idea that if we were lugging them around the country with us we better use them! So for us it was great.
We biked the closed part of the road in Grand Canyon South Rim to Hermit's Rest, an old stagecoach road in Yellowstone, the nice paved bike path in Grand Teton NP, around the campground and Lake McDonald area in Glacier NP, a rail-trail in Missoula, Mt., etc. One of our favorite memories of the entire 30-day trip is getting up early and biking the canyon road in Zion NP. Simply amazing, and totally uncrowded (we saw maybe 10 people max in an area that gets inundated during the main part of the day). And, of course, bikes are great at the campground and it's nice to be able to leave the vehicle parked and ride to the ice cream stand or whatever. You just need to look: you'll find plenty of places to do bike rides, even in parks or areas that might not seem too conducive to biking at first glance (like Grand Canyon).
Regarding transporting the trailer, that can be a bit of a pain, but I have seen waterproof bags that allow you to pack the trailer in them and then attach to an arm-type bike rack (hitch rack or similar). Space is always at a premium when RVing.