Unfortunately I cannot see the wiring diagram clearly on the side of the switch so I cannot reference the CT1 and R1 reliably.

Perhaps the following will be useful to you.
This is actually a relay and not a switch.
When you push the switch you force the relay contacts closed.

This starts the unit but also energizes the relay coil so that the contacts stay closed when you release the switch.

Looking at other wiring diagrams for units that use this setup.
A pink/black wire goes to one side of the relay coil. I think this will be CT1 on the relay.
A Blue wire goes to a common connection for the other side of the relay coil and one side of the switch. This goes to the thermal fuse.
Another Blue wire goes to the other side of the switch. This then goes to the timer, BU connection.

So if you measure Blue to Blue it should be infinite ohms and switch to 0 ohms when the switch is pushed.

I have a bad motor or timer?
It is possible.

On the timer you could check that the BK/Bu contacts close when in a run mode. Also on some unit the other side of the motor gets power through another set of relay contacts.

Wish I had a wiring diagram for this unit it would get rid of a lot of guessing.

For the motor you would have to check the winding continuity.
One thing to note is that the motor does have an intenal thermal cut-off. That is common to both windings.

If the clicking you heard was the relay pulling in then odds are that the start relay is OK.
The reason I say this is that the relay coil and the unit's motor both get power through the relay contacts. So if the relay energizes the motor should be getting power.

Here are your parts
Replacement parts for WHIRLPOOL TEDS840JQ2 | AppliancePartsPros.com
Have you checked the thermal fuse (Item 7 in Section 3)?

[COLOR="Blue"]I installed the new start relay and still no start. I did a resistance check on the timer. I read online I should get a resistance reading of 2K-3K Ohms at the main terminals with the timer in the run/normal position. I had 0 resistance. So I guess the timer is shot?[/COLOR]
If by the "main terminals" you are referring to the motor winding then you may have just measured it wrong. Since the timer motor runs and advances the timer motor is OK.
On how to measure:
Disconnect one wire going to the motor and measure across it.
Use the meter scale just higher than what you expect to see, in this case the 20K scale (20,000 ohms).
If the scale is too high it would show 0 ohms.
If the scale is too low it would show infinite ohms.

[COLOR="Blue"]If I plug the dryer in and reconnect the wires I hear the timer running and the timer advances but I guess this doesn't mean much. Correct?[/COLOR]
No it does mean that the motor is OK.
In any case a bad motor is not relevant to your problem.
It is the timer contacts that are important.
Even with a blown timer motor if you manually set the timer to a run mode the machine should start up. It just would not advance the timer.

I did some searching and was able to find the following, it at least gives the wiring for the motor.
http://appliancejunk.com/forums/index.php?topic=1430.0

[COLOR="Red"]You did not say if you had checked the thermal fuse on the blower.
Has this been checked ???[/COLOR]

Using the wiring diagram as a guide.
Unplug the machine.
Set the timer at mid point of a timed dry cycle.
Unplug the wire at BK then measure the BK/BU timer contacts.
Use a low scale (200 ohms).
It should be 0 ohms.
If OK replace the wire.
Unplug the wire to either T or W (timer switch 4) and measure the T/W timer contacts.
They also should be 0 ohms.
If OK reconnect the wire.
If both the above measure OK then the timer contacts are probably OK.
The following gets a little more complicated as we have to remove the start relay's coil from the circuit to get accurate readings.
It must be taken out of circuit because it is wired in parallel with the motor so if left in you will read the relay coil resistance.
Disconnect the pink/black wire from the start switch.
Then one meter lead is attached to this wire the other meter lead goes on a blue wire at the start switch.
And measure this at your most sensitive meter scale,
Should be 1 or 2 ohms, theses are the motor windings.
Note: That you may have 2 blue wires at the start relay one will read all the time, the other just when the start is pushed.

Basically here is the circuit path you are checking.
PINK/Black wire : Tan wire : TIMER SWITCH 4, T/W timer contacts : WHITE wire : motor's START and RUN windings : thermal fuse : BLUE wire

Hope I explained this clearly and that it helps.