One of the major updates the DSi brings to the DS platform is full network connectivity. While featured only minimally in the original DS and DSLite, download content and firmware updates are at the core of the DSi experience.
You'll figure this out as soon as you first power up the system. Click on the DSi Shop icon from the main menu, and you're immediately prompted to run a firmware update.
The process for doing firmware updates is pretty much the same as it is on the Wii. For the non-Wii owners out there, that means pretty much hands-off, although a bit on the slow side.
As with the firmware updates, the DSi shopping experience is quite similar to that of the Wii, right down to the interface. After you select the DSi Shop icon from the menu screen, you're taken to a greeting screen which will presumably be used to list promotions, notices, and latest content in the future. From here, you can select to go to the main shopping interface, which currently includes a single category for DSi Ware software (no Virtual Console category yet!).