I don't have a schematic, but the likely scenario is-

Your solenoid should have 2 small posts in the activation circuit.
One is 12V from the Key switch S terminal.
The other goes to the blade safety switch.
Blade must be OFF to complete the circuit and thus the ground.

The RED wire would connect to the + battery cable terminal and supplies the key switch B terminal.

Just one small post on the solenoid, 2 big posts. Sorry I wasn't more clear, and I didn't mention the 2 big red wires. Big red battery wire goes to one big post, big red starter wire goes to the other big post. It's the 3 little wires that I'm not sure where they go.

Looking at it again, I see that the seat switch wire is a yellow wire that goes in parallel with the black wire to the blade switch area after meeting up with the black wire. Could they be part of the same circuit? The small red wire is also coming from the key switch area, and I believe that is the key hot wire. Not sure what the green wire is, but in all likelihood, it's for the headlights. If it is, then would I want the headlight wire to go to the control post or to the same post as key hot? And would the blade switch black wire go to the control post? I think it's the green and black wires I have mucked up.

Thanks for your help! The schematic I found is a later model and uses slightly different colors, so I'm not sure of those last 2 small wires.

The crank circuit would typically go like this-
From the S(olenoid) terminal of the key switch, it would go through 2 safety switches.
The attachment/PTO switch and the brake/clutch switch.
Then on to the small terminal of the solenoid.
The RED wire to the key switch goes to the B(attery) terminal of the key switch as the "supply".

Likely, the 2 safety switches are 2 switches in one.
1/2 for the cranking circuit and 1/2 safety circuit to kill spark if out of the seat and the blade is engaged.

The cranking part-
PTO must be off to close that switch.
Clutch/brake must be depressed to close that part.

The S terminal of the key switch is "hot", ONLY in the start position.
The small RED wire between the solenoid to the key switch should have a fuse. It is also the charging wire back to the battery when the engine is running.
The only reason it's connected to the solenoid instead of directly to the battery is convenience.

Thanks! Between your points and the correct wiring diagram, I figured out where the green wire was supposed to go (ground at the solenoid mount). After replacing the fuse which I blew during testing, I now get power to the starter post with key hot, and 12v at the starter. However, the starter doesn't spin. It did spin a couple days ago during testing. If it's a bit hinky, will hitting it with a hammer help? :) Does the starter get ground from its mounting bolts? It did get caught in a recent rain and maybe a contact point rusted.

I connected my booster battery directly to the starter and it fired up (caught me by surprise!). My 12v test light came on at the starter, so unless the starter has some sort of grounding issue, I'm stumped. I'll have my wife turn it over with the key so I can test the actual voltage. The test light was one thing, but my voltmeter needs an extra hand. :)

Hah - apparently the 12v test light doesn't need 12v. :P The starter post on the solenoid had close to 12v, but the connection at the starter had less than 0.5v. I removed the battery for overnight charging, as well as the starter wire for cleaning and testing tomorrow.

Charging the battery is probably a good idea.
Also, cold temperatures can pretty much wipe out a battery that was nearing the end of its life. You might have it load tested. Places like car Quest etc. will often do this for free in hopes of selling you a new one.

IF the battery was very low, you may show 12V until the battery is put under load of the starter motor.

The battery is just off the shelf dated 10/14, and after overnight charging, showed a little over 13v. Cleaned both ends of the starter wire and the starter post looked clean and shiny. Still only getting 0.6v on the starter and exactly 12.0v at the solenoid starter post. What the heck???

I'm a bit thick-headed about how I might do it, but how can I test the starter wire itself? I don't have a substitute wire of appropriate gauge to use on the tractor.

Batteries can be bad in 2 scenarios. New & used.

Try jumping 12V to the SMALL stud on the solenoid.
It should crank the engine if battery to starter is good.
At least the solenoid should click, if it's good.

If solenoid checks out-
Using the test light-
Hook one lead to the + stud on the solenoid.
In progressive steps, connect to and then try to start with the key switch-

The "other" large solenoid stud.
Light should go out IF solenoid is good.

Starter motor stud.
Does light go out or not?

I did find a small set of jumper cables and jumped the solenoid starter stud to the starter stud. Still nothing but a loud click from the solenoid. I'm not sure that rules the positive connection out, but it seems to me there's a good chance it's the grounding on the starter? I did earlier test a battery on the starter directly using part of the starter body as the ground, and the starter turned. The new solenoid has already checked out on a bench test, so I'm fairly certain it's good. The old solenoid failed the same test.

I guess I'll try taking the starter off the tractor and see how clean the bolt holes are (assuming that's where the ground is coming from).

Does the solenoid CLICK from the key switch?
IF not, you still have a problem in that part of the circuit.

Since the starter motor cranked when jumped directly, we'll assume the battery is good.

A bad solenoid can click, but not pass the current to the starter motor.

What were the results of jumping the test light across the solenoid studs?

The idea behind the test light-
The bulb has "some" resistance.

IF it is placed across a switch/connection, a GOOD switch/connection,will have a lower resistance and the current will travel through that INSTEAD of the test light and thus the test light will go out.

I don't quite understand the bulb test.

With no key on, the bulb lights up on the battery post of the solenoid. Nothing changes to the light if I turn the key on.

With no key on, bulb is not lit on the starter post of the solenoid. With key on, the bulb lights up on the starter post of the solenoid. The small post on the solenoid which completes the safety circuit exhibits the same behavior as the starter post on the solenoid.

With key on, the starter stud test light comes on.

I'm not sure if that's the expected result, but I never got the test light to go off when touched to the battery post of the solenoid.

And regarding the battery directly connected to the starter motor, it was my car booster battery that jumped it, not the tractor battery. But again, the tractor battery is new and was fully charged to over 13v this morning.

OK, thinking about it a little bit more, I've been using the body of the tractor for my test light ground rather than the starter body. When I just tried using the starter body as the ground, the test light does not come on. It has to be a ground issue with the starter, since it fired up with me using the body for the ground with the battery directly connected to it.

I'm looking at the wiring diagram and I don't understand what a stator is, but I see a red wire with a diode on it coming out of the stator labelled red DC output and a black wire labelled black DC output. The stator is grounded. So if the stator is not grounded properly, or there is a problem with the DC output wires, could that be my issue? Is the stator the starter or some sort of alternator? It looks like a ring of magnets or something.

I ask the questions in a certain order.
Perform the tests in the same order.

1.
Does the solenoid click when the key is in the START position.
Y/N?

IF Y ignore 2.
2. Does it click when 12V is jumpered to the small terminal?
Y/N?

Thanks for your patience. Yes, the solenoid clicks with key on.

I was just now able to fire up the tractor with a jumper cable running from the body of the starter to the body of the tractor. Without that cable in place, solenoid just clicks. Somehow my ground is messed up to the starter.

The stator that I described above is somewhere in the motor and I won't easily be able to access it, if that's where the problem lies.

However, the 2 wires that run up there (black and white, connecting to the red and black DC output wires from the stator) both come from the starter switch. The black wire is on the same circuit as the fused wire to the solenoid. The white wire connects separately to the key switch. Is there a way you can think to test that circuit?

Thanks again for hanging in there. :)

Another detail... The hot wire to the stator is indeed hot with key off, but when I turn the key on, my test light turns off. Going back to what you said about the resistance in the test light, is the electricity going another route at that point? Sorry if I'm slow to get this...

Quit worrying about the STATOR!
It's in the charging system and doesn't have a thing to do with this cranking issue.

I asked 1.5 simple questions and get a dissertation in reply.
You're making it near impossible to keep things straight-

Make sure the engine & starter mounting bolts are tight in addition to the battery ground cable.

I added the extra info because of the expected turnaround time in getting replies to my posts on any forum, and thought it could be helpful.

Battery ground is clean and tight. If I ground the starter body to the body of the tractor, it cranks with the key.

I'll look to see if I can find the engine mounting bolts. The starter mounting bolts are a large star pattern that I'm not sure I have a driver for.

Have you sorted this? I read thru this post and am curious as to how you got on....

Yes, but it's not very interesting, and I'd hate to ramble on. :P

Since everything related to solenoid seemed to be working properly except that it wouldn't engage the starter, I was thinking maybe my ground at the starter was bad, I added a second ground from the body of the starter back to where the battery ground bolted to the tractor. Same result. Just a click. But a very hot ground wire. ?? I actually gave up and just decided to jump the starter with a car booster battery, mowed the lawn, and parked it. When I started it again just from the key switch and the tractor battery, it fired right up.

My guess is that the little bit of rain we had maybe froze up the engine just enough that the tiny tractor battery couldn't provide enough CCA to start it. The car battery was able to, and once it had run for a while, was back to normal operating condition.

So in short, bad solenoid and perhaps slightly seized engine that I was able to jump with a car battery.

Thanks to Bill for all his help, even though I think I gave him a headache.

"....I was thinking maybe my ground at the starter was bad, I added a second ground from the body of the starter back to where the battery ground bolted to the tractor. Same result. Just a click. But a very hot ground wire"

A very hot ground wire is your RED FLAG!
You have an insufficient ground-

Did you ever tighten the engine mounting bolts?
That grounds the engine to the frame.

The "other" important ground connection is the neg. battery cable to the frame.

Is the ground cable attached to the engine? If not it should be and a 10 or 12ga ground wire attached from the engine to the frame. This from my understanding of the OP's description.
Mike