Welcome.

First things first: there is still no way to play backup copies of 3DS games on anything. Unless that was a typo and you meant 3D games (for the DS) in which case there is nothing to worry about.
If you don't mind purchasing 2 kits (one for slot-1 and one for slot-2), go with an EZ Flash 3in1 for the GBA games and a SuperCard DSONEi (Mini) for the DS games.
If you want just one card, the SuperCard DSTWO is the way to go - keep in mind, however, that its GBA emulation is not perfect, but most games work pretty well.

(Also, update your PSP! There's like a bajillion new firmwares for it :p)