A rear-drive or 4-wheel drive car has at least one propeller shaft to the rear wheels, and some 4 wheel drive systems have propeller shafts to both axles. If you hold the shaft on each side of a universal joint, and try to twist it in both directions, there should be no "free play," which would indicate the joint is worn and should be replaced. If you turn the propshaft clockwise and counterclockwise, at the point where it enters the transmission or transfer case, there should be no free play between the propshaft and its mating shaft. Any looseness indicates the slip yoke at the end of the propshaft may be worn and need replacement.
If a propshaft is very long, it may be supported in the middle by a center bearing assembly. Check for a loose mount to the chassis, and turn the propshaft to feel for any roughness in the bearing assembly.
You should always remove any mud or debris packed into the universal joints ("U-joints"), and you should lubricate the U-joint grease fitting, if its not a sealed unit. Most 2-wheel drive OEM shafts have sealed and packed joints, on 4 wheel drive vehicles, the OEM joints have grease fittings.
An automatic transmission should get fluid refilled and the filter replaced every two years at least every two years, unless there's a strainer that's regularly serviced. For convenience, typically the filter, pan gasket and any O-rings are packaged together.
On most cars, you need to drop the oil pan to drain the old fluid and gain access to the filter. Many filters have an O-ring seal with the oil pump inlet (sometimes the O-ring is on the filter or on a neck that projects from the pump) which you need to replace it and seat it accurately. Otherwise, if the pump doesn't get enough oil, transmission performance is reduced. If there's a magnet in the drain pan (to catch metal shavings in the fluid), clean it before reinstalling.
You can also get an aftermarket in-line filter, which works like an oil filter, that you can splice into the cooler lines. If you plan heavy-duty use of your vehicle, like towing, an auxiliary cooler may a good idea, as well. If this is placed in the line ahead of the stock radiator cooler, then the Automatic Transmission Fluid isn't overcooled, and the radiator will actually warm the transmission fluid to the proper temperature in winter.