What's my best bet with a new Lithium-Ion laptop battery?

For one thing, its lithium powered.

If anything, it’s Lithium-Ion powered. Li =/= Li+

Any good laptop has a built-in charging routine. It needs to know the level of full charge and empty. Think of it like setting zero and span for an analyzer.
The manual will tell you what it prefers.

If you can find one with a higher mAh rating, select that one. It will give you longer use when running on the battery.

Charging fully before use doesn’t take terribly long and it does help a bit, so just wait the extra hour or so. I can fully charge mine from stone-dead to 100% in about three hours so just be patient.

However, once you reach 100%, there is absolutely no benefit to further charging. In fact, the light turning green signifies that it is no longer charging at all! At 100%, the charger cuts out. If it didn’t then the battery would melt. I know: I’ve done it to NiCads and lead-acid batteries as well. If you try to go past 100% then all you get are waste heat and/or noxious fumes, so LiON batteries have circuitry in them that tell the charger, “Hey, jackass! I’m full!” and then the charger stops cramming electrons in there.

As for that first discharge, I set the power options to put my computer to sleep at 3% and run it straight until it its that wall and automatically turns itself off. Actually, I do that about once a month just to recalibrate and condition stuff.

Turn your backlight down, set your hard drive to spin down quickly, and don’t use your optical drive (if you have one) when you are on battery power since that bad boy will suck down the juice real quick.