For a mailing newbie, I'd suggest getting a bubble mailer that will fit the game and then going to the post office to mail it first class with delivery confirmation. Perhaps you have a bubble mailer you can recycle?
You address the mailer just like a big envelope, your return addy in the upper left and the recipient's address in the middle.
I skip insurance for small things, but realize that if the package is mauled in transit you will probably have to cover it. (This rarely happens in my experience, especially with mail grades above media mail.)
Do get delivery confirmation, which runs $0.75 or so now. You can save a lot on this by printing your label via PayPal shipping, but I think it's best to go to the post office at first to get your feet wet.
If you can't work out how to put on the delivery confirmation label at the post office, just go up to the clerk and tell them you want to mail your package first class mail with delivery confirmation. They'll walk you through it. With retail DC I think you'd be looking at somewhere between $2.50 and $3.00 for postage.
The only other thing that may trip you up is that packages must be 3/4" thick to go 1st class mail w/ delivery confirmation. In my experience a game in a bubble mailer is 3/4" but I've had USPS clerks question it. (It's more likely they won't really care, but it does happen.) Just ask them to get a ruler and remind them that it's 3/4" thick at the thickest point of the package. I had one clerk measure the thickness at the mailer's seam, which is obviously about half the full thickness.

At any rate this will probably not be a concern, but just be prepared to clarify for them if they give you trouble.