Is there any discernable property of simple groups that can be used to quickly verify whether a group is simple other than to blast through trying to find normal subgroups?
It will become much easier to tell if a group is simple or not once you learn Sylow-theory. Sylow-theory ("see-low theory") is really just an awesomely powerful theorem, and it's primary use is to prove whether or not all groups of a certain order are simple (at least in my experience).
Other than that, I know of no way to simply tell if a group is simple or not. Obvioiusly any abelian group with a proper non-trivial subgroup is not simple.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."