Start with a cold car.
Undo the plastic tap/plug on the bottom of the radiator, have a bucket ready, or park over a drain, either way, I'm sure you don't want to have to work in a puddle of coolant! Take the radiator cap off, undo the plug, and drain the radiator.
Once the flow stops, do the tap/plug back up. On the radiator, you'll see 2 fat hoses on the top and bottom, secured with either hoseclamps or hoseclips. Unscrew the bottom first, and gently twist the hose to loosen, also it may help to have the bucket under it, as it may still have some coolant in the hose. Repeat with the top, undo the hoseclamp, gently twist it off the radiator pipe. Don't pull the hose without twisting, as its' easy to crack endtanks, most being plastic and brittle, especially with high-km cars.
Undo the electrical plugs for the radiator fan/fans, and there may also be a coolant temp sensor plug in it too, unplug it.
Undo the bolts on the top or side mounting brackets, and remove them. The radiator should be floating around unsecured now, most just sit in rubber mounts at the bottom. Have a good look, make sure you've unplugged everything, and gently lift the radiator out.
If the new radiator hasn't got the fans, transfer everything over to it, so that it ends up identical to the one you've removed.
Place gently back in car. Do mounting bolts up. Plug fans/temp sensor back in. Gently work hoses back onto radiator, I like to rub a bit of RTV around the inside of the hose for a good seal, but you don't need to, just make sure the hoseclamps are tight. Fill the radiator with coolant, watch for leaks.
Once radiator is full, leave the cap off, start the car, and run it for 10 seconds. Top radiator back up, and put cap on. Start car, run for 5-10 minutes, or until warm, keeping an eye on the temp gauge, and checking for leaks from the hoseclamps. Let cool completely (squeeze top hose to make sure there isn't any pressure left), open cap, and if it needs it, top back up with coolant. Check radiator overflow is filled up to "full" line also.
And you're done.

If you want to be thorough and do preventative maintenance, it would pay to replace both hoses while you're at it, and perhaps the thermostat, especially if the car has been running pretty hot.
Daily driver: Toyota RunX/Toyota Caldina
Ex: 2x AE101, 5x KP60, KP61, EP71, 3x KE70, KE72, AE70, AE82, 2x TE71, AE90, AE92, ST170, plus 11 Hondas, 12 Nissans, 6 Fords, 4 Mazdas, 3 Mitsis, an Isuzu and a Lada!
