Each FF game is standalone for the most part (unless it's a sequel and will then include the same number. FFX-2 for example). FF 1-6 do exist and some will swear to their greatness (6 specifically) though I started in PS1 era (7-9). 11 and 14 are MMORPGS but very much beloved by their userbase.
There are similarities between the games throughout the series but each has it's own unique battle system, world, characters, plot etc. So 7 has really nothing to do with 16 at all except for a few things such as the same summons that often appear from FF to FF, certain magics might as well, and usually Chocobos are in the game in some form (a fluffy horse sized bird you can ride). Certain names (Cid) will also be used in numerous games though the characters have nothing to do wtih each other and are often extremely different personality types from game to game. Cid in FF8 (my personal favorite due to the love story and humor found throughout the game) for example was a nerdy school headmaster and in 16 Cid was a smooth talking handsome OP rebel leader.
FF7 was probably the game that propelled the series to the forefront of gaming and helped redefine the JRPG genre, so Sqare-Enix is remaking it, but also changing it up a bit. They've split the original FF7 to 3 games in the remake, the first named Remake, second named Rebirth (coming out soon), and the last will have some sort of 'Re' in it as well. It has a more action oriented battle system compared to the old one being completely turnbased, they're also messing a bit with the plot and adding backstory to characters, but keeping the essence of the game mostly intact.