3DS games DO NOT work on DS / DS Lite / DSi. Different cartridge shape (tab on side), different CPU (ARM11 vs ARM9/7), different firmware — physically and logically incompatible.
DS games WORK on 3DS / 3DS XL / 2DS / New 3DS / New 2DS XL. Backward compatibility is native:
If you want 3DS games, you need a 3DS family system. Used 2DS ~$60–$80; New 3DS XL ~$130–$180.