If battery is more than two years of age, and has been left to sit over the winter/ non-mowing season/ without being charged, it may be too old to be used this year. Also, use new cable bolts at the battery connections, and the one where the gound cable is bolted to the frame or sheet metal by the battery. And a good cleaning of the connections is warranted, too.
From your description, it would seem that the grounding of the "minus" cable, or corroded cable to battery connections are the main problem.
Also, if the engine is the over-head valve type, the valves may be out of adjustment. Since you have professed a lack of mechanical bent, you may want to ask somebody else in your area. But, i would first check the battery connections, etc.
HTH: RustyJ

This has all the classic symptoms of the engine valves needing to be adjusted.
Proper valve clearance is essential for the Briggs compression release (decompressor) to function effectively permitting the engine to spin up to cranking speed quickly when you first turn the key.
I'm a little surprised you have not had to deal with this already..............it being an '03 model.
Briggs says to do this annually. Most folks do it "as needed".
If you are in no way mechanically inclined.......you had best farm this out to your local small engine shop.
It is not a very complicated (for them) procedure and the only parts involved might be a valve cover gasket that is not fit for re-use.
The fact you can start it OK after you have run it a while likely means your battery is just a bit stronger shortly after the recharge it gets in use.

Thanks for the quick replies, guys. Sounds like whatever it is it's neither fatal, nor all that expensive to correct, which is good news.

To clarify just a bit, the battery is less than two months old, put in by a dealer. So how would I go about cleaning the connections, etc.?

We're in peak growing season right now, and yard is well fed, so I don't want to take it in right away. Am I risking more serious damage by using it, as is? And am I looking a a tractor that eventually just won't start until the problem is corrected (if it is the valve adjustment thing)?

Thanks again.

Continued use (regarding start up only) of the tractor while this condition persists will certainly hasten the wear to the brushes in the starter, and will stress the battery more than it would otherwise.
How much sooner the starter will wear out or the battery will wear out is anybody's guess.
You might want to modify your "starting technique" a bit.
Instead of simply holding the key in the START position until the engine finally begins to spin, try this.
And you must react quickly.
Turn the key to START. If the engine does not turn, release the key and quickly turn the key to START again. If the engine fails to spin, repeat the process until the engine DOES spin.
This way you will catch the engine in a part of the 4 stroke cycle that has little or no compression to overcome, allowing the engine to "get spinning" before it comes up against the next compression stroke.
When you just hold the key in START and the engine is not spinning........you are burning the brushes in the starter and overheating the starter armature too.