IF the battery hasn't been load tested, it's suspect.
What method was used to test?

To test the solenoid, simply jump across the 2 large terminals. IF it cranks, the solenoid is bad.

IF the battery & solenoid is good, then you have a bad connection in the crank CIRCUIT.
This includes GROUNDS, which includes things like engine mounting bolts.
Clean ALL cable connections on BOTH ends.

Thanks. I took the battery in and had it tested. Brand new...don't think that's the problem. Solenoid is new but I'll test it too. The solenoid has two connections....one white and one black. I connected those two but there is an additional white wire not connected and I don't see where this one is supposed to be connected to.....it's just sort of free floating out there. Is this a ground wire you are talking about?

BATTERY cables are usually RED+ & BLACK-

Typically on a Craftsman, the WHITE wire goes to the small terminal on the solenoid.
It provides 12V from the key switch to "activate" the solenoid.
The solenoid is simply a switch that connects the battery + to the starter motor, to spin it. (it does go through the clutch/brake & attachment switches)
Otherwise, you'd need a super heavy duty key switch and long heavy cables that went from the battery to the key switch to the starter motor, with the voltage drop disadvantages of long cable runs.

IF the WHITE wire is disconnected, the solenoid shouldn't be clicking.
Are you sure the clicking you hear isn't the carb fuel solenoid? The starter solenoid should make a "HEAVY" click sound.

You can get a schematic at-
searspartsdirect.com

This post was edited by bill_kapaun on Sat, May 17, 14 at 16:25

Yeah I've got a small white and a small black wire connected to the posts on the solenoid. There is a third white wire however in that bundle and it's not connected to anything. That's the one I'm wondering about. The wire is not long enough to stretch too far so I'm thinking that if it's to be connected to something it should be rather local and I'm just not seeing what that could be.....

The solenoid has to have a minimum of 3 terminals.
2 heavy ones to carry the current for the starter motor from the battery and 1 to activate the solenoid.
Some solenoids have a 4th small terminal for a ground (BLACK), while the 3 terminal ones are grounded by the case.
You have access to the schematic!

This post was edited by bill_kapaun on Sat, May 17, 14 at 23:41

Unless that "unaccounted for" White wire is in the vicinity of the clutch/brake switch, or of the PTO switch, I would not be too concerned about it.
I don't know where your extra White wire is on your tractor, but unless it is near the clutch/brake switch, or the PTO interlock switch for the deck, it is likely an extra wire the OEM included in that harness for use with a different model tractor.
OEMs of all sorts of vehicles use the practice of including "extra" wiring in a harness in order for it to fit several variants of a basic model.
This keeps them from having to make up specific harnesses to fit each individual model coming off the assembly line.
One thing about the harnesses used on Craftsman tractors is that no matter what the model, or year built.....the White wires always function for the starter solenoid control circuit.
So, you need to locate the PTO interlock switch, and the clutch/brake interlock switch.......and see if you have an empty terminal on either switch. There are 2 White wires on each of those switches. There are also 2 Black wires on each of those switches.

Thanks so much for your help Bill and Mownie. I appreciate it. I did some checking yesterday and I believe my trouble is in the starter. I've got juice getting to it but it's not cranking. So....I've ordered a part and we'll see what happens. In the meantime I'm checking into the PTO interlock switch and clutch/brake interlock switch.....

Did you clean all the heavy cable connections?
DID you jumper the 2 large studs of the solenoid?

Something else not considered yet-
Remove the spark plugs and see if the engine cranks.
Possibly the carburetor leaked through and hydro locked the cylinders. IF so, the oil would have a distinct gasoline smell.

Do you have a voltmeter? IF so, simple voltage drop tests can tell you where the problem lies.

You still haven't been real clear on HOW the battery was tested.
Some think that if you have 12V, it's good. 8 penlight batteries will generate 12V, but won't have the CAPACITY to crank an engine.

This post was edited by bill_kapaun on Sun, May 18, 14 at 12:43