There have been many iterations of the Pi since its 2012 launch, but to enjoy superior performance from Kodi, OSMC, or OpenElec, use the most recent model, the Raspberry Pi 3B+.
For a standard setup, relying on the microSD card is fine. However, if you plan to use the Raspberry Pi as your Kodi system long-term, then a more reliable storage solution is needed. The Raspberry Pi supports USB storage, which you can use for your own library as well as downloaded media.
Connecting a USB HDD to your Raspberry Pi for Kodi is a smart idea. Keep reading for more on this, and some other advanced hardware options.