The elevator to the Big Room leaves from within the visitor center, and the walk from the visitor center to the natural entrance of the Big Cave is very short. Most of the trail in the Big Room is wheelchair-accessible. Flat-soled shoes are strongly recommended for the hike into the cave from the natural entrance, and a good idea if entering the cave via the elevator. Running shoes particularly cross-trainers or light hiking boots work well.
Many publications assert that the temperature in the Big Cave is "a constant 56 degrees F," but the truth is a little more complicated. The temperature varies from place to place within the cave, and atmospheric conditions can create small temperature variations in some locations as well. However, 56 degrees F 13 degrees C is a good number for planning purposes. Take a light jacket or wrap into the cave, even if it's beastly hot outdoors.
If planning to do any backcountry hiking, make sure to take plenty of water and sunscreen; there is precious little shade, and no reliable water sources. Serious hiking boots are a very good idea, as the terrain is rough, plants are thorny, and rattlesnakes though rarely encountered typically strike at the ankle level. In the backcountry, long pants are a better idea than shorts, if you can stand the heat; a little protection from spines and thorns goes a long way.