Small Engines (Lawn Mowers, etc.): Lawn Mower, starter solenoid, sparkplug

Question
Michael, thanks for the help.  However, when I went out to lube the starter, the mower wouldn't even turn.  There was no response at all.  So, I jumped the mower and it started up and then I mowed my lawn.  That is problem one...I can't get the mower to start without jumping it.  Now I have a new battery so I don't really know where to go from there.  

Additionally, after running the mower for about an hour, I heard what sounded like a backfire and then the thing died.  So I don't really know what to do now.  Thanks again for your help and if you have any insight on these problems I would greatly appreciate it.  

Matt

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Followup To
Question -
Michael, I have read through some of your previous answers and while there was a ton of info, I don't think it hits my specific problem.  I was given a 1990 Craftsman 2 Riding Lawn Mower.  Last year it worked fine.  However, this year when I went to start it up, nothing happened at all.  So, I replaced the battery and sparkplug.  

Now, when I turn the key, there is a buzzing noise but it still doesn't start.  I pulled the whole engine apart and see that when I turn the key, the starter/solenoid turns but does not catch the flywheel.  It spins below the flywheel so it doesn't even turn the engine over.  To be honest, I don't know a lot about small engines but figure with a 1990 model, I either fix it myself or trash it as I do not have a way to transport it to get fixed.  Additionally, I don't know if I am using the right terminology.  Thanks in advance.  
Answer -
Hi Matt,

If the starter is spinning normally and fast, the gear should rise and engage the flywheel.  Sometimes it's spiral shaft gets dry and dirty.  Try cleaning it and spraying it with a non-dirt-attracting lube, like teflon or graphite.

Let me know how you make out.
Michael

Answer
Hi Matt,

Maybe the starter was not spinning fast enough to engage the gear.  Go through my No Cranking proceedure below, using what applies, to check out the starting system.  Clean all the connections to bare metal.
The backfire may have been due to a safety switch cutting off the ignition or a faulty key switch.

Keep me posted.
Michael
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I would check the starter first.  Jump the positive battery terminal to the
starter and it should crank over. If so, skip to step 2.
If not, jump from a battery that you know is good, from a car.  If still
nothing, the starter is bad.  If good, then the mower battery is weak.  

Step 2.
If the starter and battery are good, you will need to trace the entire circuit
from the starter to the battery.   The solenoid, switches the battery power to
the starter, when power from the key switch is sent to the small terminal on the
solenoid. To test the solenoid, first make sure that you have power at its large
terminal which comes from the battery. Then jump power from that terminal or the
battery positive post, to its small terminal. You should hear a click and the
other large terminal that feeds the starter should be energized. If not, then
the solenoid is bad. If good, then continue tracing the power through the key
switch to the battery until you find the break in the power path.   All testing
can be done with a 12 volt light or a voltmeter.  Most equipment has safety
switches in  the seat and on the blade engagement lever. Check them by
temporarily bypassing with a small jumper wire.