You really need to post the full, technical model number of the tractor in order for us to check online for a wiring diagram in case it comes down to checking specific components and circuits. The model number I refer to will usually be found on a decal under the tilt up seat. Typically it might begin with 917. xxxxxx.

But your problem could be caused by a poor connection where the battery negative cable fastens to the chassis or engine.
Ground path is just as critical as the positive cable circuit.
There might also be some kind of "parasitic drain" being imposed on the battery (though I would have though a new key switch should have eliminated that).
Post the technical model number and we will try to go from there with things for you to check. You will want a volt meter (tester) to carry out the checks.

Some good suggestions from Mownie. Problems such as yours require skills with a test light or test meter. Frequently, battery connections or ground connections to the frame are tight, however there is still resistance in the connection. Boosting the battery may increase voltage just enought to overcome the resistance and the engine cranks. Reply with results.

Since the starter solenoid clicks when you turn the key to the start position, check the voltage delivered to the starter terminal. When performing this check position the common lead of the voltmeter against the tractor frame.

Model Number is 917.273821

Engine is 31P777-0296-E1

Yes, 31P777. We now need to consider 2 things that could be affecting this problem. First, when the engine does its "crank a bit" act...........do you mean it acts like it wants to crank but just moves slightly and then stops turning?
If you wait a minute it will repeat that? and it might finally catch on and run, sometimes?

If that is what it is doing, I suggest that the engine is probably past due for its recommended annual valve adjustment. Have the valves ever been adjusted in the time you have owned the tractor?

But, if the action you are seeing/hearing produces a "rapid fire clicking noise" from the starter solenoid for a few tries and then finally does nothing............I suspect that the battery is not being recharged when the tractor is operated. If recharging with a battery charger, or boosting from a good battery produces a dependable cranking, that sort of points toward a problem in the engine charging system. I see that the tractor has a manually engaged PTO system. If there was an electric PTO clutch you would also be complaining that the deck would not stay engaged, and that would be a symptom to weigh in on the non charging scenario.

It is real easy to determine whether the charging output is OK.
All you need to do is read the battery voltage while the engine IS NOT running and write down the voltage on paper. Battery voltage at this point should read at least 12.3 volts and might even read 13 volts if it has been charged recently.
Next, start the engine and run it at full throttle. After running for a minute (and while it's still running at high RPM) check the battery voltage again and write down the voltage on paper. The voltage should read at least 13.7 volts and hopefully as high as 14.3 or so.
If you do not observe at higher voltage at max RPM compared to the not running battery voltage, your charging system is defective.

It acts like your first description exactly and no it has not had a valve job in the 6 years. So I found some advice on how to adjust the valves, tried it and screwed it up so bad, not sure where I am. I understand the locking nut and screw, but not sure I had piston in right location. The instruction I saw said to turn engine until intake valve was closed, adjust intake to .oo4 in, then exhaust to .004. Can you provide better instruction on how to set this up, now that I have it totally screwed up or should I just take to a tech. I would like to be able to do this myself in the future.

If you will send me an e-mail (use the member profile feature to see my address) I will send you some literature that will straighten you out.
And be careful to use the correct terminology here. What you need is a simple "valve adjustment", not a full blown "valve job". Even if you decide to seek a tech shop after reading my offered tutorial, you don't want to say "valve job" casually.

I have same problem. Battery is ok but need to jump with car battery to start. Even then I have to position the fly wheel past the compression point in order to gain momentum to push through when I turn the key. Plus backfires sometime when I shut off. Its give years old and I just bought in the spring. Started ok then but steadily worse over summer. Can you send me the same literature? Thanks.

Can you send me an e-mail? It is in my member's profile.