Thanks to one of our good friends, we've officially rung up our first online sale!  Last night, we sold a Black-Purple/Gold Gemini Polyhedral 7-Die Set from Chessix.  Hooray!
Surprisingly, all the backend stuff seems to be working out for the most part.  The online store is hosted by Square, and that's tied into our QuickBooks Online account.  The only oddity that I haven't figured out yet is that the Square Fees aren't being imported into QuickBooks, so the transaction looks slightly higher than it should be.  Since the import is on demand, I'm going to wait to see if, perhaps, the fee gets properly tacked onto the transaction when the funds are deposited in our bank account.
We're still trying to figure out our Point of Sale solution in the store.  Square is convenient, but it's rather pricey on transactions above $22.95.  Also, they don't handle cost of goods at all, don't track suppliers, and have a bit of an odd process for managing employees (as near as we can tell, each employee would have to sign up for Square unless you want all your transactions to be, essentially, anonymous).
For the moment, we're using ShopKeep as our primary POS, and Square for our online presence.  If we can figure out a way to manage the inventory properly, it might make sense to use Square in the store for transactions below the $22.95 threshold, and ShopKeep for everything else.
For those wondering, Square is cheaper at the low-end is they use a flat rate of 2.75% per transaction.  Other credit card processors charge a lower percentage, but typically have a per-transaction fee ($0.10 – $0.20).  With ShopKeep, you can use any processor that allows for integration.  We've looked at Payment Revolution (recommended by ShopKeep) and Chase Paymentech (they recommended ShopKeep to us originally.  Awkward.)
Payment Revolution seems to have the lower cost offering ($0.10 per transaction, and their rate is only 0.25% above interchange), and they don't have any setup fees.  Of course “interchange” is a nebulous number – it varies by brand, card type, etc.  Square keeps their pricing uniform, which makes it easy on our side of the house, but, for larger transactions in particular, you're paying a premium for that.
But, regardless of that, we have our first sale!  Woo Hoo!