I start with castile soap or Belvoir glycerine, on a very slightly damp sponge. Once a week I condition with the Belvoir Leather Balsam Intensive conditioner, although it is pretty heavy duty stuff. Our tack room is very dry and the leather often needs more moisture than if the tack room were slightly more humid. For really really nasty tack that has a lot of built-up sweat/dirt/dander on it, I use the Belvoir "Step 1" spray, which cuts right through the grime but definitely demands that you use conditioner afterwards.

I like to take my bridles apart every time I clean them... I just find it easier that way, although I know a lot of people prefer to leave them buckled up. Usually I will clean and condition the bridle, and while the conditioner is soaking in, I clean the saddle/girth/leathers/irons. By that time, the bridle is ready to be wiped down and buffed up.

I clean my tack at least once a week, usually more. I'm responsible for many sets of schoolhorse tack at the barn where I work. The riders do a quick wipe-down after their lessons, but the real cleaning is left to the grooms. I love cleaning tack; I find it really relaxing and it makes me feel like I've accomplished something when I see a nice row of gleaming bridles and saddles.