Do I need to "train" my new battery?

Yes, typically you should train it.

Please see this article:

http://support.apple.com/kb/ht1490

or follow these steps:

PowerBook G4 (15-inch Double-Layer SD), MacBook (all models), MacBook Pro (all models), MacBook Pro (17-inch) (all models) and MacBook Air

The battery calibration for the PowerBook G4 (15-inch Double-Layer SD) and any model of MacBook or MacBook Pro has been updated because of a new battery released with this computer. With these computers, follow these steps to calibrate your battery:

1. Plug in the power adapter and fully charge your PowerBook's battery until the light ring or LED on the power adapter plug changes to green and the onscreen meter in the menu bar indicates that the battery is fully charged.
2. Allow the battery to rest in the fully charged state for at least two hours. You may use your computer during this time as long as the adapter is plugged in.
3. Disconnect the power adapter while the computer still on and start running the computer off battery power. You may use your computer during this time. When your battery gets low, the low battery warning dialog appears on the screen.
4. At this point, save your work. Continue to use your computer; when the battery gets very low, the computer will automatically go to sleep.
5. Turn off the computer or allow it to sleep for five hours or more.
6. Connect the power adapter and leave it connected until the battery is fully charged again.

Here is one of the best resources on battey management:

https://discussions.apple.com/thread/1764220?threadID=1764220

It's a user-contributed tip that's been well-refereed and vetted before posting. It's like "one-stop shopping" for maintaining Mac notebook batteries.