I'm waiting for my Panasonic GH3 to arrive from B&H. When it does, is it better to charge up the battery first before using?
It won't hurt, but it'll also be okay to just use it. Lithium batteries (like that used by this camera) last longest when stored at around 40% charge, so hopefully that's what they do before they put them in the box. So, it's probably fine to take it out and play right away for a while, but you won't get as long of a first trial period as if you charge it fully first.
Many people find it valuable to have a second battery. I recommend buying one right with your new camera purchase. Then, you can start using one while you charge up the other.
If you're intending to do a firmware upgrade, do the full charge before starting that.