Many of Yellowstone's most popular areas are easily accessible via paved trails and boardwalks -- this list includes the Old Faithful area as well as other geyser basins, Mammoth Hot Springs, and the Lower Falls at the Grand Canyon of Yellowstone. You should have no problems with strollers in those locations.  Please note, however, that the boardwalks can be slippery if near geothermal activity, so be extra careful!

One good hike I can suggest is the Back Basin Trail in the Norris Geyser Basin.  It's a 1.5-mile loop that passes Emerald Spring, Steamboat Geyser, Cistern Spring and Echinus Geyser. 

For more great walks and hikes click here:  (http://www.ohranger.com/yellowstone/walking-hiking)

Have a great trip!