Without power, can you spin the motor shaft by hand? Does it spin free or is it hard to turn?
Can you smell any burning smells if you put your nose near the motor? If that motor hums when you power it and it does not run it could be a damaged bearing, my reason for asking if the shaft spins free. Bearings in some motors are easy to replace, but some of this type motor do not lend themselves to easy replacement.
Are there brushes in this motor? If so there should be 2 round plastic caps located at 180 deg opposite to each other on the motor sides. These caps can be removed to access the brushes, but be careful since there is a strong spring beneath the cap. You don't want to loose either the spring or the cap. The brush under each cap and behind the spring will be a black carbon rod with either square or round sides. If the brush is very short (less than 3/8" long) they need replacing. If the brush is broken or missing, especially if the end of the spring is burned, the motor is likely damaged and also history.
If the motor windings have been overloaded and burned, you should be able to smell the strong pungent burn smell when placing your nose near the motor, and if you can smell that, the motor has likely already had the magic smoke let out of it, and it's history. A new motor will be necessary.
If you connected the wires to the two left or the two right terminals on your new switch and there was a flash when you turned the switch on, did the flash come from the motor or the switch? Is there still power at the outlet that the saw was plugged into? Can you plug something else into that outlet to see if it has power?
Accurate answers to these questions is the only way that we can help you without being there to do it for you.
In reviewing your predicament ... your table saw was working, then it died in the middle of use, and then a friend of yours suggested buying a new switch. After your struggles in figuring out how to connect the new switch, you reconnected the old switch again, and after doing so, a change in state of the saw occurred. The saw hummed. The fact that the saw hummed means that the old switch functioned well enough to pass power through it, which indicates that the root cause of your problem wasn't likely the switch after all.
The thing is, with a continuity tester, as Woodnthings suggested, you can not only test the new switch to verify for yourself which terminals make or break when the switch is on or off, you can also test the old switch too (without it being connected to wall power). Having that capability to test can help you demonstrate to your friend that the problem isn't the switch.
Is there a lot of sawdust under the table of your saw? An excess accumulation under the saw is often accompanied by an excess accumulation of sawdust inside the motor. An electrical motor packed with sawdust can overheat quickly, become finicky, or even catch fire. If you haven't done so already, consider vacuuming out the cabinet of the saw, and then, simultaneously with the vacuum in operation, directing a nozzle of compressed air through the openings in the motor to expel the dust accumulated inside. Be sure and use a different outlet, feed by a different circuit in the home or shop, to segregate where the vacuum cleaner and the air compressor are plugged in, to avoid running two high current loads on the same circuit simultaneously for prolonged periods of time. It might take time too. It took me as long as a full hour to clean all the dust out of one table saw and motor that I acquired used.
With your switch function confirmed and better understood by you, and with your saw motor completely clear of dust, and your saw underpinnings now clean enough for you to locate and access the components on the motor, such as the thermal reset switch on the motor itself, the capacitor cover if the motor is so configured, and the shaft, to test for restriction in rotation by hand (when disconnected to power)... you are now ready to try all of Charley's suggestions quoted above, which are all excellent questions for you to answer, and a recommended path toward diagnosis.