Tank Size Count on 10 gallons per inch of turtle as a general rule of thumb, and I recommend a 20 gallon tank as a minimum for hatchlings. Keeping in mind that red eared sliders can read 10-12 inches as adults, you can see that eventually you will need a very large tank. You do not need to use an aquarium though -- large plastic containers or storage tubs are fine as long as you don't mind not being able to view the turtles from the side. As long as the tank is tall enough and the basking area is positioned so that the turtles can't climb out you won't need a lid.
Red eared sliders need an adequate amount of water in which to swim. At a minimum, the water should be at least 1.5 to 2 times as deep as your turtle is long (so a 4 inch turtle should have a minimum depth of 6 inches). The length of the water area should be at least 5 times the length of the turtle, and the width should be at least 3 times the length of the turtle. As long as your turtle can get out of the water and there is no place your turtle can get trapped underwater, you do not need to worry about your turtle drowning -- they are strong swimmers.
The basking area can be provided by stacking smooth rocks, sloping smooth large gravel to one side to make a land area, using wood, or a plastic "turtle dock." Whatever you choose to construct a basking spot, make sure your turtle can climb onto it easily, and that it allows your turtle to completely dry off.
When designing a tank, keep it easy to clean (i.e. uncluttered). Keep in mind that turtles may knock things over, and push them around. Plants may be a nice touch, but turtles are likely to make a snack of them or uproot them (plastic plants will likely be uprooted and just make cleaning more difficult). Likewise, gravel can be a nice touch, but can make the tank harder to clean and you must make sure it is large enough that it won't be accidentally swallowed.
Between feeding and defecating, turtles are pretty messy creatures. A turtle tank should include a good filtration system such as a power filter or canister filter. Choose a filter rated for at lease double the amount of water you will be filtering. Filtration will reduce the frequency of water changes, but your turtles will still require 25% water changes weekly and a thorough cleaning once a month or more. Feeding your turtle in a separate container helps minimize the mess in the tank and reduces the load on the filtration system.
The water should be kept at 75-78 F (23-26 C), and 78-80 F (26-27 C) for hatchlings.
The ambient air in the tank should be around 75-80 F (24-27 C), with a basking spot at 90-95 F (32-35 C)
A submersible heater can be used to keep the water heated if necessary, but a broken heater is a potentially lethal situation. You can protect the heater so the turtles can't bump it by placing it behind something (e.g. bricks) or by fashioning some sort of cover. (A reader submitted this suggestion for a heater guard: place the heater inside a section of PVC pipe. Choose a piece with a diameter significantly wider than the heater and a bit longer than the heater, and drill many holes in the sides of the pipe to allow water circulation past the heater.) You can also get durable plastic heaters such as the Visitherm Stealth heater (see one here). Get a good aquarium thermometer (not a stick on strip) and monitor the water temperature.
A heated basking spot should also be provided over the area where the turtle can get out of the water. A reptile bulb/heat lamp in a reflective cover can serve this purpose, but make sure there is no way the turtle can touch the light or that the light can fall into the water. The basking light will help heat the air in the tank, as well.
In addition to the basking light, a full spectrum reptile UVA/UVB light should be provided. Exposure to UVA/UVB is necessary for proper calcium metabolism, and also appears to have other benefits to overall health such as improving appetite. It is also nice to take your turtle out into natural sunlight in warmer weather. Get a proper reptile UVA/UVB bulb, and follow the manufacturer's guide regarding placement of the bulb (UV rays diminish the farther away you get from the bulb so placing the bulb in such a way that the turtle can benefit from the UV is important). It is also important to replace the bulb as recommended by the manufacturer, since the intensity of UV produced diminishes over time.