Indoor:
First and foremost, you will need a large container that will hold water. If you have a baby snapper, a 20 gal tank (preferably 20 gal long) will suffice for a few years. As your turtle grows, you will have to increase the size of his tank. When in the market for a tank, take into consideration that your turtle will get bigger and bigger. A full grown snapping turtle could be kept easily in a 100 gal wide/long tank. Since your turtle will need room to swim and turn around, any tank you get should be of the wide or long variety (vs tall). You can actually save money by getting a large tank to start off with. That way you don't have to keep replacing tanks. And no, they don't get bigger because they are in a larger tank. If the size of environment determined the turtle's size, we would have house sized turtles swimming around in the Great Lakes.
How deep should the water be? The water should be at least as deep as the turtle is wide, but shallow enough for the turtle to rest on the bottom and still reach the surface by extending its neck. Unlike most other aquatic turtles, snapping turtles spend most of their time on the bottom. If you force your turtle to swim to breathe, it may drown. If your tank is large enough, you can have deeper areas for the turtle to swim and dive in, but make sure it can't become trapped in these areas.
Your tank should be set up for easy cleaning, since you will be changing the water about once every other week. As far as what to put in the tank, I suggest large aquarium gravel for the basic strata with pieces of rock and wood. If you decide to place aquatic plants in your pet's environment, make sure they are not poisonous since they will eventually be consumed. Snapping turtles love to dig and hide, so place things in its environment that are easy to get under. Avoid sharp and jagged rocks since your turtle will cut its shell trying to get beneath these items. I have found that hollow logs cut down the middle to form a wooden 'cave' work nice. There is some disagreement over whether or not to put gravel or sand in your tank since your turtle will eventually consume either. But it would be almost cruel to not give this type of turtle something to dig in. That is what it spends a large amount of its time doing in the wild. This is why I recommend large, smooth, aquarium gravel for the strata. It gives the turtle something to dig in and is too large for a small to medium sized turtle to swallow. Look for gravel about the diameter of a penny.
You'll need to provide an out of water basking area also. Most snapping turtles won't use it very often, but you should provide one anyway. It only needs to be large enough for the turtle to get entirely out of the water on. Focus a basking light on the area and keep the light on all day. This helps your turtle regulate his temperature.
You should also provide full spectrum UVA/UVB light for your turtle. This can be a little pricey but it's needed. Buy a hood that runs the length of your tank. Most large hoods have places for 2 tube lights. The tubes usually cost in the $25 range but can be higher. This in another area where getting the large tank to start with will help save money. You'll only have to buy one hood. Plus, you'll only have to buy tubes when they burn out. The light will help keep your turtle's skin and shell healthy and it aids in digestion. One word of caution, don't get 'nocturnal' or 'black' lights. They put out too much UV light.
To keep the water clean, you will need some kind of filter. Look for a water filter that would normally be sufficient to clean at least twice the amount of water if it were for a fish tank. Turtle's produce MUCH more waste than fish and thus need more powerful filters. Also, the more powerful the filter the less often you will have to clean the tank. The filter should use several different types of media, including carbon. My current set up uses a Magnum 350 canister filter and 2 biowheels.The biowheels cascade water down a rocky area at the end of my tank. The canister filter uses filter floss to remove debris and carbon and ammonia reducing pellets to remove harmful chemicals. The biowheels add biological filtration in addition to the biological filtration of the algae on the rocks which the water runs down. This filtration set up lets me get away with cleaning my 75 gal tank only once per month.
Place your indoor turtle cage in an area that doesn't get direct sunlight and is not a high traffic area. The sunlight causes quick algae growth and the more secluded your turtle is, the less stressed it will be (remember these things don't like company).
The temperature range most snapping turtles are comfortable with is between 60 and 80 degrees F. Get a little stick on tank thermometer so you can make sure your turtle's tank is at the right temp. The higher the temperature the more activity the turtle will have but you will also have to clean the tank more often. I recommend a temperature in the mid 70's. You'll probably need some type of heater to keep the water warm in the colder months. There are several different types available including: under-tank, in-tank, and filtration line heater.
Under-tank heaters are basically like heating pads that slide under the bottom of the tank, not in the water. They are fine for small tanks but they don't work very well for tanks over 40 gal. In-tank heaters look like test tubes with an electric heater inside. This is what I use and I've had good luck with them. In-tank heaters are made of glass and can be dangerous if not installed properly. Since these turtles are very strong and move the contents of their tank around almost every night, you'll need to place the heater in a safe area. If it's broken, it will probably kill your turtle and may start a fire. Only use this type of heater if you can place it so that it will not be broken. Also, I've heard from people who have had their turtle's cooked by a stuck thermostat on one of these heaters. You can avoid this by getting a heater that is designed for the amount of water you have in the tank. Remember, the tank won't be full so use the actually amount of water in the tank when picking a heater. Filtration line heaters work very well but can be expensive. Basically they are out of tank heaters (safer) that tie into the water line from your filter to your tank. It heats the water at this point and the heated water is spread throughout the tank.
Outdoor:
Again, you must have a large container that will hold water. A small pond or fountain will work fine. Even a child sized wading pool will keep your snapper very happy. The main thing you need to worry about here is if you have a young turtle. You have to keep it separated from neighborhood dogs, cats, raccoons, bears, or what ever the heck roams through your neighborhood. A small 'chicken wire' cover will be fine for most outdoor cages. As far as worrying about your turtle wandering off, you shouldn't have to. Snapping turtles spend most, if not all, of their lives in water, only leaving to find food. So if you keep your turtle well fed and with plenty of clean water to live in, it shouldn't ever decide to take a stroll. But if you are still a little worried, place a one to two foot high fence around your turtle's outside home (don't worry about this at all unless the container is 'in ground').
How deep should the water be? There should be plenty of areas where the water is at least as deep as the turtle is wide, but shallow enough for the turtle to rest on the bottom and still reach the surface by extending it's neck. Unlike most other aquatic turtles, snapping turtles spend most of their time on the bottom. If you force your turtle to swim to breathe, it may drown. If your tank is large enough, you can have deeper areas for the turtle to swim and dive in, but make sure it can't become trapped in these areas. I
For decoration, place items you would find in a pond or stream. Pieces of wood, rocks, and aquatic plants are all good choices. If you decide to place aquatic plants in your turtle's environment, make sure they are not poisonous since they will eventually be consumed. I recommend sand or dirt as the strata for outdoor setups. You may want to consider pea gravel or large diameter aquarium gravel if you are using a plastic or concrete pool.
Filtration for outdoor setups will vary with the size of the setup. A large setup will probably not require a filter since nature will take care of the job. Smaller setups however will need some kind of filtration unit. Look in farm supply stores and nurseries to find fountain or small pool filters. They don't have to be too powerful since larger amounts of water are usually kept clean by plant life.
If you are not using a pre-existing pond or fountain, build your turtle's home in an area that is partially shaded and away from roads, paths, or driveways. If the water is deep enough, your turtle will regulate it's own temperature. The only time you really need to worry about too much sunlight is if you are using a wading pool or like sized container.
*Common Sense tip of the day:
Also, remember snapping turtles are generally not very friendly to their neighbors. If other animals are kept in the same container as your snapper, they will either be eaten or carry bite wounds. So if you thought you would have a nice little outdoor fountain filled with goldfish and a snapping turtle for your back yard, well... consider a red-eared slider.
On occasion (about once or twice a month), you will want to use either a soft vegetable brush or old toothbrush to clean your turtle's shell. This will prevent shell rot and fungus growth. Just run warm water over your snapper and lightly scrub the shell (watch your fingers, this is a situation in which you are very likely to be snapped at).
If you notice your turtle's nails getting long, use regular bird, cat, or dog nail clips (depending on the size of your turtle) to clip the nail back till the white part of the nail is almost gone. Also, watch for a small dark dot in the center of the nail. This is a blood vessel. When this appears, stop clipping. If bleeding does occur, use a styptic pencil (get it from a drug store, it's the same thing people used to use to stop shaving cuts) and press it against the nail tip till the bleeding stops. You may want to wrap your turtle in a damp towel. This will relax your turtle and lessen your chance of being bitten (which, in case you haven't figured it out , is very high in this situation). If you give your turtle something to dig in, such as rocks or sand, you won't have to worry about this.
Like birds, snapping turtles' beaks keep growing. Unlike birds, I seriously doubt you will be able to cut back the over growth. To avoid this, place a cuttle stone in the turtle's cage so it can chew on it and naturally keep it's beak trimmed back. Replace the cuttle stone when it begins to turn green.
Outdoor setups usually take care of themselves. If you notice your outdoor tank getting dirty, you can generally clean it with ordinary pool cleaning equipment. If it's small enough, use the same method as described for indoor tanks.