Official route only: The Nintendo DSi Shop closed on March 31, 2017. You can no longer purchase new DSiWare titles.
DS/DSi cartridges work natively. Buy used from eBay, GameStop, or local stores.
Install TWiLight Menu++ (via Memory Pit or Flipnote Lenny exploits) to run homebrew, emulators, and backups from an SD card. Requires SD card, PC, and following a guide (e.g., dsi.cfw.guide). This voids warranty (moot) and risks bricking if done incorrectly.
Piracy warning: Downloading ROMs/ISOs you don't own is illegal. Support developers by buying original cartridges.