What are good methods of locating plumbing blockages?
Call a pro
Call Roto-Rooter®, or any other plumber of your choice.  They'll be able to feed a camera down the line, and tell you exactly where, and what the clog is. Even without a fancy camera, a plumber should be able to use a regular snake to locate the clog.
Use a snake
If you want to give it a try yourself, plumbing snakes (auger) are available at any hardware or home improvement store.  They're available from many different manufacturers, in both hand powered, drill powered, and motor powered depending on your budget.
Using a snake is easy, but it takes a bit of practice to get a feel for locating clogs. Keep track of how much line you have fed out, and make note of it when you hit the clog.  Then measure along the pipe, to determine the location of the clog.
Magnets
If the pipes are all plastic (PVC, ABS, etc.), a strong magnet might help.

Get a strong magnet with a hole in it.
Tie a string to the magnet.
Put the magnet in the drain, and use a large volume of water to flush it down the drain.
Run another strong magnet along the outside of the pipe, until you feel the magnet inside the pipe.

Flushable transmitters
Flushable transmitters are available, that can be sent down the drain line.  Once in the line, the transmitter can be located using a receiver.
Credit to @Ed Beal
Robot
If the pipe is large enough, or you can find a robotic ball small enough (Sphero for example). You could send the robot down the drain to locate the clog. Getting the robot back, might be a challenge.
WARNINGS:

Anything you put down the drain will have to be retrieved, or small enough to be processed by the waste treatment center.
Anything put down the drain should not be damaging to the drain system, nor the sewage system.