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

Yes, I read the similar question from a week ago, and I wasn’t satisfied with the answers…

I’m expecting the battery I ordered to be at my house by the time I get home from work, and I plan on charging it as soon as I open it. This probably isn’t necessary, but it’s a 10.8V, 6600 mAh, 9 cell, Li-Ion battery manufactured by Lenmar. I’ve heard several tips on first-time battery use, and I want to get some feedback to help me maximize the battery’s charged-time.

I’ve heard to charge it up fully before use. Does this mean before I power up my laptop? Should I leave it on longer than the time needed for it to reach 100%?

I’ve also heard to let it fully discharge the first time. Does this mean I should let my battery die and let the laptop turn itself off? Or should I shut down the computer when the battery is very low?

Any other tips would also be helpful.