A stock DS lite can play Gen III, IV, and V on real carts. As the best selling of the three, there's a lot out there, but it can be hard to get one in great shape and a decent price. Lots of scratched screens, dead pixels, broken hinges and touch screens.

A stock DSi can play Gen IV and V on real carts. Easier to get one in good shape and for cheaper, because of the lack of GBA slot and general marketing toward adults and not kids.

A stock 3DS can play Gen IV, V, VI, and part of VII on real carts or digital, and Gen I and II on digital. Lots of model options though going to cost more than the others though, as it's not that old.

If you buy a DS flashcart, you can play Gen I, II, IV, and V without any problems on any of these systems.

If you buy a GBA flashcart, you can play Gen III on the DS Lite without any problems except you won't be able to transfer your Pokemon to Gen IV from the flashcart.

If you mod a DSi (free) you can play Gen I and II easily, Gen III (might have issues) and Gen IV and V (probably without issues).

If you mod a 3DS (free?), you can play Gen I and II easily, Gen III, IV, and V (probably without issues) and Gen VI and VII.

ROMs Note: I'm not advocating piracy, but no matter your feelings or experiences, be warned, if there's one thing Nintendo is the most salty about, it's Pokemon DS piracy. They still monitor that above all else.

TLDR:

Using real carts only: DS Lite if you need Gen III, 3DS if you need Gen VI and VII. DSi to save money if you only care about Gen IV and V.

Using ROMs, the absolute cheapest option: modded DSi, play Gen I - V (III might be iffy). Modding isn't hard, but you'll have to do it, or get someone to do it for you.

Using ROMs, the most possible games: modded 3DS, play Gen I - VII. Modding is harder than on the DSi.