If you're interested at all in Metal Gear Solid 5, then grab the Metal Gear Solid Legacy Collection (or just 4).
If you're interested in MGS5 then the ones you need to focus on are MGS3 Snake Eater and MGS : Peace Walker, in that order.
In chronological order:
Big Boss story arc - MGS3, MGS: PW, MGS: GZ, MGS5: TPP
Solid Snake story arc - MG1, MG2 , MGS1, MGS2, MGS4
The Big Boss story arc is tragic and heartbreaking and is far more interesting than Solid Snake.
I cried so hard when Peace Walker concluded. Dude means so well and touched so many people's lives has been through hell for no reason because other people are power tripping assholes. And it all started with one act that completely changed him and he carries the guilt and regret with him his entire life and it comes to define him and his legacy.
MGS3 and MGS PW build up Big Boss as this really good guy and MGS5 is supposed to be where he gets screwed over one last time and finally loses his shit and becomes the legendary villain we all know. (remember Sniper Wolf in MGS1 talking about how she was rescued and given a life by Big Boss? etc).
MGS3 hit the feels hard but MGS PW just crushed the feels even more showing how he still can't let go of the events in MGS3 10 years earlier, and then in MGS5 GZ we see everything dear to him (his family essentially) taken from him without provocation, which culminates in the creation of Outer Heaven and the need for nuclear weapons (Metal Gear) as a deterrent against epeen tripping super powers that don't like competition from a mean well group without borders or ideology or political allegiance.
I'm going to be hitting MGS5: TPP hard this September.
But yea, blow through MGS3 and Peace Walker back to back. It's an amazing story that will tug the feelz hard and get you ready for some vengeance in MGS5 as we see Big Boss turn to the dark side.
--
We will forsake our countries, we will leave our Mother Lands behind us and become one with this Earth.
We have no nation, no philosophy, no ideology, we go where we're needed, fighting not for country, not for Government, but for ourselves.
We need no reason to fight, we fight because we are needed.
We will be the deterrent for those with no other recourse.
We are soldiers without borders, our purpose defined by the Era we live in; we will sometimes have to sell ourselves and services.
If the time demand it, we'll be revolutionaries, criminals, terrorists.
And yes, we may all be headed straight to Hell, but what better place for us than this?
It is our only home.
Our Heaven, and our Hell.
This, is Outer Heaven.
- Big Boss aka original Snake