After you've created an account with us, you'll be prompted to input some details about your brand, specifically a preferred font, a primary and secondary color, and a logo. After that, you'll be asked to log in and authorize us to pull your business's information from Square, like your locations and catalog. Finally, you will be asked to checkout, but this is only payment authorization, no payment is captured (a technical term for actually transferred) until firstly, you have your betas to test (in the case of the publishing charge), and secondarily your app is on the store (in the case of the per-month charge). After that, your app will be ready to sell items from your catalog and push it to your Square POS!
In terms of the App Store and Google Play Store themselves, each upload is always a little different. In general, it takes under a week to go from "Marked for Release" to downloadable on your customers devices, with the Apple App Store being in general a bit slower than the Google Play Store. Don't worry though, you will not be charged the monthly fee until your app is available on both stores.