You can jump into the new one without too many problems, and when you understand that Drake can jump an inordinately long distance and has the hands of a gecko, it's fine. You probably don't miss much when it comes to starting half-way through the series - only character backgrounds and previous associations, which isn't that important.
Gameplay is a bit like Tomb-raider, but with more action. I've not played the third one all the way through, (played the second one through in two sittings on a friend's PS3) but in general there is a mixture of action, stealth and puzzle parts; there is quite a bit of time given to gunning, but you can reduce enemy numbers by stealth-KOing them, so in a way it's quite well integrated.
The game itself is very cinematic, so in a way I guess the FPS analogue would be CoD; it's mostly linear in a similar way - same start and end, but you can get to them by different means, depending on where you are. It's not very long, probably between 5-10 hours for the first playthrough, but it's very well done.
If you're not sure, it might be worth looking up a longplay of the previous ones on youtube, as the gameplay and mechanics through the series is pretty much the same.
Personally, I thought #2 was a bit on the short side (which is par for the course these days), but otherwise very well done. The bits of #3 that I played were also pretty good.