Metal Gear: Portable Ops (2006)
(Naked) Snake is called out to hunt his former unit, FOXHOUND, and chase Nazi gold or something in South America. Introduces a lot of characters who become important later, and explains why they're all so psychologically knotted up in each other and Snake, as well as where the man himself will get some of his crazy ideas. Oldest canonical appearance of a Metal Gear. It's also a PSP game, so not many people played it.
Metal Gear: Peace Walker (2010)
More secret agent shit with nukes happens in South America again. The man who was Naked Snake fully assumes the mantle of "Big Boss", and by the end of the story, embarks on his quest to save... the world? Save soldiers? Big Boss's obsession with soldiers and countries of mercenaries and stuff is supposed to start to make sense somewhere around here, but since it's another PSP game, I don't know anything about it. Supposed to be pretty good.
Portable Ops is apparently no longer part of the continuity/canon in favour of Peace Walker, since it wasn't a Kojima written game.
Peace Walker is a pretty good game, I actually started a topic about it on these forums somewhere. It's gameplay is probably most directly related to Portable Ops, but since I haven't played that, I find it plays like Monster Hunter with guns and stealth mechanics. It also can be really bloody hard in some parts for no good reason, but that is probably more indicative of control problems (Your thumbs are not designed for the level of control the game requires sometimes) rather than gameplay problems.
It's also not as 'funny' as any of the other Metal Gear Solid games. There are codec calls, and there are some funny moments, but the game is played really, really straight. It's perhaps played too straight some times during the story, but side/secret missions make up for it somewhat.
(And then it gets into creepy/weird territory) There's a demo of Peace Walker that you can download to your PSP, it's probably worth a shot (if you have a PSP and enjoy Metal Gear games)
A whiny Big Boss, resentful of the US gov't after its behaviour in MGS3, has taken some loyal soldiers and is operating as a mercenary corp in South America. He's hired by a Costa Rican politician to investigate a US presence. As in all Metal Gear games, this presence happens to be nuclear equipped special weapons. He then mans the hell up, kills and tranqs thousands of soldiers, saves the world, and becomes a complete badass with his own private nation where everyone loves him. Really the plot kind of dies after the whole world-saving bit but it's supposed to tie together Metal Gear and Metal Gear Solid 3. I remember it doing so I just can't remember how or why.
Large spoilers below!
The special weapons are commanded by artificial intelligences that are voiced by vocaloids. You can make your own vocaloid.
The end-boss AI is "The Boss" from MGS3 so obviously Big Boss remains super sappy throughout. Otacon's dad got plans from his buddy Granin in MGS3 that inspired him to build the end-boss. He has a walking wheelchair. He also makes you a Metal Gear.
After saving the world, Big Boss then goes on dates with his best buddy and a very young girl.
An antagonist is named Dr. Strangelove.
The main bad guy is named Hot Coldman and is not from a MegaMan game.
You can meet a talking cat.
You can recruit Hideo Kojima.
I can't believe this. I actually can't buttonspam enough to survive Ocelot.
I had a similar problem in Peace Walker for... oh, 6 months. You'll figure it out eventually, I'm sure!