Occasionally you may need to transport a pet red-eared slider turtle by vehicle, e.g. if you are moving. This can be a very stressful time with turtles, but with some planning and preparation, the move can go quite smoothly.
For travel in the car, your turtle can be placed in a plastic box such as a storage container. Holes can be drilled in the lid of such a box for ventilation, but you will want something that closes securely since turtles can be adept at escaping. Place a good layer of padding in the bottom of the box, such as a towel (but avoid any with loose threads or loops that could entangle the turtle's claws), paper towels or newspaper. The box should have a bit of space for the turtle, but don't give the turtle lots of room to move around (the goal is to pad the turtle well and make him or her very secure).
What about Water?
Even semi-aquatic turtles such as red-eared sliders can handle being out of the water for quite a while. For extended trips, you may choose to mist the turtle and or dampen the padding at the bottom, but keep in mind this will be cooling to the turtle so only do this if you can keep the turtle warm enough. For trips of a few hours, your red-eared slider will be fine if kept dry. However, on longer trips, you might want to take a break and offer your turtle some water to drink.
This will be your biggest challenge, though typically overheating is probably more of a concern that being too cool (though a cold turtle will be sluggish). Use a thermometer on the turtle's box to monitor the temperatures. If you are traveling in warm weather, there are a few alternatives. You can regularly mist your turtle or keep the padding damp; evaporation of this water will be cooling. If your car has air conditioning, you can use this to keep the car cool, being careful not to let the turtle get too cool (and don't put the turtle in front of an air vent). You can also place the turtle's box inside a cooler (the styrofoam ones work well for this) and add a wrapped cool pack to the cooler. Never put an ice pack or cool pack directly in the turtle's box—keep the cool pack separated from the turtle itself.
In cooler weather, you may need to use a warming pack in much the same way. Place the turtle in its box inside a cooler, and add a wrapped heat pack (a bottle of hot water, or a chemical heat pack for longer trips). Again, never let the turtle come in direct contact with the hot pack.
If you have drilled some holes in your box for ventilation, your turtle will probably be fine. However, especially for longer trips, you may want to open up the box periodically to let in some fresh air.
Since turtles can quite readily go a couple of days without food, you can take your turtle off food for 24 hours before your trip (and the duration of the trip, unless it is longer than 1 or 2 days). This will greatly reduce the chance of your turtle soiling its box.
Overnighting in Hotels
If you need to stop overnight, make arrangements well ahead of time and make sure you get permission to keep your turtle in your room. Bring a medium-sized plastic storage tub—you can set this inside the bathtub and give your turtle some warm water to swim in. Be courteous to the cleaning staff and other guests by not putting your turtle directly in the tub or sink or letting it roam around the room (due to the risk of contaminating surfaces with Salmonella bacteria). When you pack your turtle back up in the morning, simply dump the turtle's water down the toilet. To be extra courteous, wipe down the bathroom surfaces and anywhere else the turtle may have had contact with anti-bacterial wipes.
A note on plane travel: the source linked above give some information and advice on traveling by plane with your turtle. See the links above for more information on preparing for air travel.