Well, the answer to your question depends. With the Kobo, you will need to set up your Kobo account. After that, if you want to buy books from Kobo, it's just as simple as with your wife's Kindle Fire. Browse the store on the reader, buy it with a few clicks, it downloads to the device and you're ready to read.
If you want to read books from other stores then Kobo's, it can get slightly more complicated. Most other stores (like Sony's) sell epubs with DRM by Adobe Systems. So to handle books like that (or library books from Overdrive for example) you would have to install Adobe's Digital Editions software and set it up to work with your Kobo.
You are definitely not required to use Calibre. However, as it will manage both your Kobo-bought books and side-loaded books, a lot of people find using it is to their advantage.
Hope this helps a bit. I'm sure more eloquent folks then I will be along shortly...