There are typically three main aspects to a successful surgical procedure---the pre-surgical phase, the anesthesia and surgery itself, and finally, the post-operative period.
The pre-operative phase entails the pre-surgical diagnostics, surgical preparation, and pre-anesthetic procedures. The anesthesia and surgery is fairly self explanatory, and are, obviously, extremely important. But, it is important to note just how critical the post-operative period is to the entire process. Actually, successful procedures can fail simply because of mistakes or problems during this critical period. I've seen animals who were healing and recovering beautifully regress because of not following the post operative instructions given to its parents.
The basic key to success during this time is to follow all instructions. Most of what is recommended during the post operative period is common sense-minimal exercise, short controlled leash walks (assuming any walking is allowed), keeping the incision lines and/or the bandages or wraps clean and dry, not allowing the pet to lick or bite at the incision line-use that Elizabethan collar-it looks funny, but it's effective, and continuing to administer medications per the doctor's instructions.
Some fairly universal instructions are: sutures should usually stay in for 10 to 14 days before removal. For surgeries done using either laser or electro-surgery, we usually recommend they stay in for at least 14 days, and sometimes even a little longer. Bandages, wraps, and casts should, ideally, be checked weekly. If you note any swelling, irritation, moisture, or odor around, or emanating from the wrap or cast, make sure to have your pet rechecked immediately. If, despite good care and medication, your pet seems to be in excessive pain or discomfort, have him or her rechecked as well.
If you follow your post-operative care instructions to the tee, your pet should heal well and recover beautifully. Make sure to give your pet the best chance of having its surgery be a success!