I would ignore remakes and remasters and play them as released, MGS3 is pretty well positioned to satisfy some of the questions and lore exposed in the previous two games, which is not as satisfying/memorable if you play it as an intro.

MG1 and 2 do contain quite a bit of lore but it's ok to skip them if you're not big into retrogaming. Both NES MG1 and twin snakes are mostly cash grabs that Kojima wasn't involved in, the former only makes sense if you have trouble with MSX2 emulators and can deal with a worse port, while TS is just non-canon.

A small note: almost every game makes some "creative use" of the original hardware - discs, controllers, etc - which don't translate very well to emulation. Of course they don't impact the story and they are limited to a few moments, with one exception - you should really get a DualShock and an adapter if you can, as the game does use the analog face buttons quite a bit. Most notably, if you want to lower your weapon without shooting after aiming, you need to slowly release the square button. If you use almost any other controller that becomes impossible as no matter how slowly you go it will be read as 1->0 and it will shoot.