Release order. MGS2 is the best one. MGS3 has a very different kind of story, but is a refinement of the MGS2 play system, and has its own compelling story.

MGS4 mechanically feels really good, but interrupts its own gameplay too often to show you a cutscene. When you can play it, it's fun. The story gets worse (stupider) from this game and onward.

Peace Walker is a pretty fun game. I enjoyed it on PSP, but there's no doubt that game feels better to play with additional buttons from the HD version. It's literally more ergonomic that way.

MGS5:GZ is a highly re-playable and fun sand box and sets up its successor nicely. MGS5:TPP is an incomplete game that takes too long to beat and under-delivers on what was promised. My excitement and subsequent disappointment over it stopped me from ever pre-ordering a game again, and mostly got me out of AAA gaming.

-- How to play

Emulate as much as you can. If you have a pressure-sensitive DualShock 2 or DualShock 3 you can use those in PCSX2 for MGS2 and MGS3. Otherwise, like Gorb said, try the Xbox360 or Master Collection. Otherwise you'll do unintentional things on the inputs, such as never being able to grab someone in MGS3 without immediately killing them. Some people like the pressure sensitive buttons. I think they're kind of a poor use of the feature, but it is what it is.

Peace Walker emulates nicely on RPCS3, but MGS4 is still a little troublesome with stability and performance. You can still try it, nevertheless.