I wash my gloves and pack in the washer when I feel they need it. I don’t like to do it too often, as the washer seems to tear them up quicker. I just spray off my shoes with the hose if they need it, and I never wash my helmet. I have changed the pads in them with the spares that come with the helmet when you buy it.
Keep in mind I’m from California, so I never see mud. That might change my washing habits.
Fabrics, get the washing machine; Camelback, gloves (only sometimes, because they wear quickly out of a washer), 5.10’s etc.
Helmets; handwash as I did the same when I raced quads, seemed to work best.
Goggles; handwash as above
Clip shoes; same as cavermatthew said, toothbrush and a rag
Cleaning armor/gear is a cinch. Spray it off with the water hose and be done. Shoes get the soapy rag and sprayed off. The inside of my shoes get sprayed with antifungal spray and once they are older they get submerged in a antimicrobial bath solution and then oiled and left in the sun to dry. If your shoes are not leather do not oil them. My knee braces are scrubbed with soapy warm water using a brush then layed out to dry inside by a window or heating unit if it is winter time. My shorts and shirts are washed after every ride in the wash machine (not recommended with chamois but I am lazy) as are my socks. I replace chamois every 3-6 months. I ride every day usually for around an hour or so.
get a mesh washer bag and throw your gloves, chamois, helmet liners right in there. The bag protects the Velcro from itself and other things from it, and just seems to keep things contained nicely. Then I can throw my shorts and socks and calf sleeves etc in the same load and use special synthetic-fabric detergent. Don’t use fabric softener.
My gloves go in the washer with my shorts, socks and other clothes. Once every few weeks I dunk my elbow and shin pads in a gallon bucket filled with warm water, a cup of white vinegar and a couple tablespoons of baking soda. It completely takes the stink out of them. I wash my Osprey pack using dish soap, by hand in the sink.