I would say start from the beginning and work you way towards the latest installments that way you would see how the series progressed.
If you have a Game Boy Advance you would be able to play: Final Fantasy, Final Fantasy II (both are on one cartridge known as Final Fantasy I & II: Dawn of Souls), Final Fantasy IV Advance, Final Fantasy V Advance, & Final Fantasy VI Advance.
The PSP has Final Fantasy: Anniversary Edition, Final Fantasy II: Anniversary Edition, Crisis Core: Final Fantasy VII ( a prequel to FFVII that is more an action-rpg), & Final Fantasy Tactics: The War of the Lions which is an update of Final Fantasy Tactics.
With a DS that can play Game Boy Advance games you can play all those that I listed for the Game Boy Advance plus: Final Fantasy III, Final Fantasy IV DS, Final Fantasy XII: Revenant Wings.
The PS2, with backwards compatibility, can play Final Fantasy VII, Final Fantasy VIII, Final Fantasy IX, Final Fantasy X, Final Fantasy X-2, Final Fantasy XI (a MMO), Final Fantasy XII, and a spin-off SRPG - Final Fantasy Tactics. Also, there are a few compilations of the older games: Final Fantasy Origins (FF & FFII), Final Fantasy Chronicles (FFIV & Chrono Trigger), & Final Fantasy Anthologies (FFV & FFVI). Though I would stay away from these compilations because the GBA or DS version* (* that apply) are better versions with some bonus content.
The PS3 and Xbox 360 both can play Final Fantasy XIII.
A good thing to keep in mind about the series is that each game is different from each other. There are some themes that carry over from each iteration but don't expect to the next game in the series to be like its predecessor. The stories are not connected so you don't need to play Final Fantasy II to understand Final Fantasy V for example.
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