I realize this is a lot of reading, so I've just tried to condense the ideas down into easier to read tips below.
You may notice that some of the tips below often involve listening to what your cat is telling you about what she prefers in terms of the type of box and litter.
Starting your cat off right by choosing the right box, the right litter, and the right placement will help to prevent possible problems in the future.
Note: If your cat shows any sudden change in bowel or urinary habits, sleeping, eating, or drinking, or personality and behavior it could be a cat illness warning sign. You should report any of these changes to your veterinarian.
I was looking around for some decent videos on cat litter box training, and I have to say, there are some pretty lame ones out there. In any case, there are some good ideas in the video below, but I wouldn't normally do "Step 5" where they recommend moving your cat's paws through the litter.
The Dumb Friends League recommends against this, and I have to say that I think some cats would think you're an idiot for swishing their paws in the litter. Cats are much smarter than people and they know what to do with the box. There I said it, cats are smarter than people.
Litter Box Type, Number, and Placement
Cat litter box training begins with the right selection of the type, number, and placement of boxes.
In multi-cat households, follow the "one plus one" rule. One box for each cat in the house, plus one extra. This ensures that a free box is always available, and prevents disputes due to box guarding.
Some cats prefer one box for urination, and one for defecation -or- they won't use a recently used box, even if they were the one to use it. So, the "one plus one" rule may apply to houses with single cats as well.
Hooded litter boxes, also referred to as covered litter boxes, tend to reduce the amount of litter scatter, as well as reduce the odors that get into the air. They'll also cut down on "litter dust" so that may be important if you have allergies (like me). They also tend to look better since they visually hide the litter and anything in it. Be aware, however, that some cats may decline to use a covered box since the odors may be more concentrated inside the box.
If your cat refuses a covered box, you can opt for one of the cat litter furniture options on the market. These come in various designs. Some of them have better venting than others, so make sure you either choose one that seems well vented, or make a vent for any that you buy. That way, both you and your cat will be happy with your choice.
Place litter boxes on hard, flat surfaces, like tile, for added box stability. Some cats will avoid boxes placed on carpet or soft surfaces as the plastic box may move too much as the cat moves around inside the box. Your cat needs to feel safe in there.
The box should be placed in a quiet area (but one frequented by your cat), away from food and water bowls, and with multiple escape routes.
A box should be placed on every floor of a multi-story house for easy access. You can't place a litter box in the basement where your cat never goes, and expect positive results.
Many adult cats will prefer a larger box (meaning wide so they can move around). Some will like boxes with higher sides, some lower.
Kittens and smaller cats will need litter boxes with lower sides. A lower, more easily accessible box will keep your kitty coming back.
Cats with mobility problems, injured or older cats with arthritis, or cats recovering from surgery may need special access. A high litter box may be a problem for these cats. Make accommodations.
To give access to a higher box, "pet stairs" products can be used. These are often used to give access to beds or other places a pet with mobility problems might need to get to. Be aware, though, that a high box may be hard to get out of, so a cat with mobility problems may simply need a lower box.
Type, Brand, and Amount of Litter
The type, brand, and amount of cat litter may important to your cat. All the right moves in terms of cat litter box training will be of no use if your cat doesn't like the texture or scent of the litter. Some cats are pickier than others, and your cat reserves the right to become picky at any time!
Most cats would prefer slightly more litter in the box over less. Typically, 3 to 4 inches of litter is better than the often seen 2 inch recommendation. Your cat may have her own preference.
Some cats (especially medium to long hairs) prefer to move the litter aside and defecate on the hard box floor. For these cats, slightly less litter is often preferred. If your cat does this, one box with more litter for urination, and one box with less for defecation may be your best option.
Cats normally like loose, sandy materials as a bathroom, scooped at least once per day. Alternative litters may work (flushable cat litter, pine, crystals, Cat Genie granules, non-clumping litters, etc.), but generally, scoopable unscented clumping clay litter seems to cause the fewest problems for cats.
Your Cat Wants Choice, and Slow Changes
Whenever possible, let your cat make the choice as to which box location or type, or which kind of litter (or how much). Cats tend to resist change, and prefer a routine. Make changes slowly.
Find out how much litter your cat prefers: placing two side by side cat boxes with differing amounts of the same brand and type of litter will tell you how much litter your cat prefers. Whichever one your cat uses most over a week's time, switch to that.
Find out which type of litter your cat prefers: two side by side boxes with different types of litter will do the trick. Whichever gets used the most is the winner.
When switching litters, do it slowly. Mix in about 20% of the new litter with the old. Over the period of a few days or a week, add in more and more of the new litter until it's 100% new. If you first do the side by side comparison, you'll know before hand whether or not your cat will accept the new litter without problems.
When introducing a new litter box, leave the old one in place and allow your cat to choose which one to use. If your cat takes well to the new box, then you have a winner. This can be especially important when switching from a manual type box to an automatic litter box. Some cats may be afraid to tackle the automatic boxes, so let your cat get used to it and make the choice.
Litter Box Aversion and Box Maintenance
Once the initial cat litter box training is over, and you've gotten all the decisions right, your cat shouldn't have any problems. Still, inappropriate elimination, or litter box aversion or avoidance may happen at some point in your cat's life.
If, in the absence of any stress, life changes, external factors, and despite a meticulously clean cat litter box, your cat suddenly starts urinating outside the litter box, you should assume a medical problem is at fault. Male cats with urinary tract problems can be facing a potentially fatal medical emergency.
Scoop your clumping litter at least once a day, preferably twice. Keeping the box as clean as possible helps to prevent box aversion problems.
Keep litter levels fairly constant by replacing litter as needed. Replace the litter entirely, and thoroughly clean the box every few weeks. Use a 1 - 20 bleach solution to kill germs, and rinse and dry thoroughly. Do not leave any residual bleach smell or your cat may avoid the box.
Use a comfortable litter mat. Most cats don't like the prickly feel of the pointy fake grass type.
Let's say your cat litter box training goes successfully, but later your cat develops a litter box problem. Remember that most of these problems are temporary. Many litter box problems come down to either medical problems, or some sort of reaction to stress of some kind. Addressing the medical problem, removing the stress, and retraining should fix the problem.