Are all the Nintendo DS series the same for games?

don't know what you mean by "new ds games" but as long is it's a ds of any kind, it should play them. but if the cartridge says 3ds or is one of the few games that's for dsi (unless you're playing on a dsi), then it can't play them

New 3DS Games can only be played on New 3DS/2DS systems.

3DS games can only be played on 3DS/2DS and New 3DS/2DS systems.

DS games can only be played on DS, 3DS/2DS and New 3DS/2DS systems.

So on and so forth.

Quick List:

DS: DS and DSi enhanced games

DS Lite: DS, DSi enhanced, GBA games

DSi: DS and DSi enhanced games

Old 3DS: DS, DSi enhanced, 3DS games

Old 2DS: DS, DSi enhanced, 3DS games

New 3DS: DS, DSi enhanced, 3DS, New 3DS games

New 2DS: DS, DSi enhanced, 3DS, New 3DS games

If you're asking about older cartridges in a fresh console, they should work perfectly.

Long Version:

The DS, DS Lite, and DSi can play all DS games. 

The DS and DS Lite can also play "DSi enhanced" games, just lacking the camera feature that comes with playing on the DSi.

The DS Lite, and only the DS Lite, is the only DS console that is backwards compatible with GBA games.

The DS, DS Lite, and DSi cannot play 3DS or New 3DS games.

The 3DS and 2DS systems can play all DS and 3DS games.

The old 3DS and 2DS models cannot play "New 3DS" games, there are only four of these: Fire Emblem Warriors, Minecraft, Xenoblade Chronicles 3D, and Runbow Pocket Deluxe Edition.

New 3DS games can only be played on 3DS/2DS models with "New" in the name.