If you're looking for the kind of storyline you described and you don't mind clunky controls and old game mechanics you should start off with MGS1 and go chronologically. If you don't care about storyline and you want a wonderful and tense stealth experience you should give MGS5 a try.

MGS5 with the HUD and the radar deactivated and using headphones is the most immersive and tense game I've ever played by far. Just lying down on the floor and listening to the footsteps, the smoking, the conversations of the guards around to know where they are and how to get to them stealthily and knock them out.

All MGS games can be played two ways: stealth or action. As action games they're meh, the combat mechanics are clunky and unrealistic and not that fun. The stealth part is the juicy one, try and restart missions if you get caught instead of just shooting your way in, much more rewarding and tense.