Final Fantasy 1 through 10 are brilliant. Well some are weak, but most are among the best ever games, if you enjoy long story-driven rpg adventures. I see you like Nintendo games, so you should love these titles as they have a similar level of quality as Nintendo software.
You should naturally start with FF1, which was originally published by Nintendo in the US. It establishes many of the series conventions, such as classes, menu styles, musical themes, ancient times mixed with modern (or ancient) technology, elementals and so on. Nowadays I would recommend playing one of the remakes... I enjoyed FF1 on my PSP... since the original looks pretty dated.
FFIV represents the pinnacle of early NES/SNES-style FF, with a great classical serious story, beautiful music and great refinements to gameplay that were seen in FF1 and FF3. FFVI is loved by many and begins the transition to the style of the Playstation-era games, though it is still in 2D. Its setting is a little more modern and story more convoluted than the previous series benchmark in IV. I like them both in different ways.
FFVII is great but slightly overrated because it was the most hyped entry in the series. Published by Sony as a reason to buy the PS1 instead of N64, it was the biggest seller of the series. It's also the first disc-based game and first 3D entry. It doesn't age all that well but it's still a classic. The series creator lists FFIX as his personal favorite. It's a bit retro, hearkening back to the NES days.
FFX to me represents the last entry that I would call an "improvement" on the series. The first PS2 entry has enhanced visuals, voice acting and an epic story and soundtrack. If you only like modern games, that may be the one to pick up. I think of it as a true classic, one of the best games of all time.
FFXII is the last entry I really enjoyed, but it's a step down from FFX in terms of polish, soundtrack, characters and story. It switches to real-time MMO-style combat which may or may not be an improvement.
You should note that many entries in the series were developed by a "b-team" and vary greatly in style. These titles tend to feature a "job" system where characters' skills are not set with the character, but with an interchangeable class system. These games tend to be weaker in the story department and in style. Examples of these "team 2" games are FF2, FFV, FFVIII and FFX-2. I definitely would not start the series with one of these games. The series creator and music composer left the company after FFX and there is a distinct drop off in the "magic" of the series afterward, especially the distinctive mood-setting music. They now make games for their own company, Mistwalker, so after FFX you may want to move on to Last Story on Wii. Xenoblade is also a good spiritual successor as many of the rest of the Square staff moved to Monolithsoft and made that title. I would skip the letdown that is FFXIII. For me, the creative talent is no longer there.