|Take advantage of the refracted light and do not use your flash.
Bring a tripod. If you do not have a tripod, use the walls to steady your shots (you can also shoot from the hip, but this is less stable).
Bracket your exposures and play with the shutter speeds.
Wear a jacket or sweater (even in summer).
Enjoy the place -- immerse yourself in the experience; not just taking photos.
By the way, private vehicles are not allowed, nor unescorted visitors, so you need to hire a tour guide.
Check out this tour company http://www.antelopeslotcanyon.com (take the photo tour if you can afford it; it is longer and targeted at photographers -- I took the shorter tour).
Also, visit the nearby "Lower Antelope Canyon" (need to pay and climb a bit), the "Cathedral Canyon" (you need to take a tour), the "Horseshoe Bend" (free 3/4 mile hike), and the "Rainbow Arch" (you can take a long hike or take a boat tour to it).
Antelope Canyon (18)