Yes, you will be able to play most Nintendo DS games on your Nintendo 3DS. Exceptions are games that use the GBA Slot. Note that some Nintendo DSi games bought outside the PAL region may not be playable on a Nintendo 3DS from the PAL region.
If the game supports wireless communications, you will also be able to connect to other people, whether they are playing the game on a Nintendo DS, Nintendo DSi, Nintendo DSi XL or Nintendo 3DS.
In the table below you'll find a detailed overview of which software is playable on which system.
*1 - Only Nintendo 3DS software can be played in 3D. Nintendo DS and Nintendo DSiWare software will be displayed in 2D on a Nintendo 3DS system.
*2 - Select Nintendo DS games that use accessories in the GBA Game Pak slot of the Nintendo DS system are not compatible with the Nintendo 3DS, Nintendo DSi, or Nintendo DSi XL systems.