Handling:

Do not pick up the head by the ears or muzzle. Over time this can cause the ears or muzzle to come lose.
Don't poke or allow others to poke the eyes. (Yes, some people do try to poke the eyes.) Poking the eyes will result in tearing of the eyes and/or hurting your own eyes.

While At Events:

Be careful. Your vision is somewhat limited.
Doing things like flips or climbing is a bad idea. You can injury yourself and/or damage your costume.
Avoid things like cars, escalators, messy foods, colored drinks, open flame, etc.
Do not swim in your costume. The chlorine and the weight added by water can damage your costume.

Suiting Up:

I recommend putting on the feet, tail, then head, and then the hand paws in that order. If your costume has a body, put the body on first.

Washing:

Washing machines and dryers can destroy a suit! Fake fur is made from plastic, it can melt. I recommend hand washing and/or spot cleaning and letting it air dry. Be sure to wash with cold water. Hot water makes colors fade and could melt the fur if the water is hot enough. Brush out the fur before and after it drys.
Heads: Can only be spot cleaned and the inside can be sprayed with Febreeze.
Hand and Feet Paws: Can be hand washed with soap and cold water. Air dry.
Tails: It's recommended to spot clean, but tails can be washed the same as paws.
Body: Can be cleaned just like paws. If airbrushed, it should be spot cleaned around airbrushed areas.

Brushing:

Fur can be brushed with a fine comb or pet hairbrush/slicker. Don't be rough or you will tear out fur fibers. Faux fur does shed some when new. After heavy use, areas can begin to look scruffy. Just comb/brush out these areas.

Traveling/Storage:

Store your costume in an area away from pets, high temperatures, and children.
Do not leave in direct sun light for hours. This will cause colors to fade.
It's recommended to put dryer sheets in the paws and head when storing to keep your costume smelling fresh.
Don't squeeze your costume into rather small places when storing or traveling. This can damage it.