What should I keep in mind when altering the drainage around my basement?
I've only done the french drain thing; I have no idea what would be required with tying into a storm sewer; you'd likely want to contact the local government and/or water company to ask.

Anytime you're digging, you'll want to be aware of where underground utilities are.  In the DC area, you call 'Miss Utility', and they'll mark out everything.  (I think technically, you're not required if you're digging by hand, but it's still a good idea).

You might also want to take a look at what large trees are nearby, as you might have to deal with roots -- you can cut them as you go, but that might end up killing the tree or weakening it so it sheds branches, and large branches falling on your house or car is not a good thing.  (especially when you have no power, so get trapped in your (damaged) home).

You'll want to look at what's down-slope of the area you're dealing with, and how much area you have to absorb the water once you move it down there, if you're going the french drain route.  If it's an area that already gets pretty marshy after a rain, it's not going to absorb anything, so you'll need to go far enough out 'til you find somewhere that'll take the water.  You also don't want to try to force it out near the property line, or you might piss off your neighbors (and if they're lawyers, or just happen to like suing, it could get messy).

...

And after all of that, you might want to look into if the area's abnormally wet -- one of my friends had problems for years, and dug up his yard multiple times, trying to get everything fixed, redid his basement at least twice because of water damage.  And it turned out that there was a broken water pipe next to his house that was saturating the lawn, so any significant rain couldn't be absorbed.