So I just completed Dante's Inferno, finally, and in the end actually found myself kind of enjoying it. This is even though I'm one of the people who were initially extremely offended that Visceral dared to defile and rape such a beautiful poem. Then I thought to myself, I've never actually played a God of War game, and seeing as people were calling Dante's Inferno a so-so GoW clone I figure I should check it out sometime. 
 
This is therefore my question to y'alls. Would I be missing out on a lot of important story stuff if I jump straight into God of War 3, or would it not really matter?

Nnnot really, and most of the prior happenings are explained to you through GoW 3 as well.  
 
The need to know: Kratos is a fucking cunt, who fucks up everything and everyone in his path.

Would I be missing out on a lot of important story stuff if I jump straight into God of War 3, or would it not really matter? "

Now I don't know what to believe. 
 
Anyway I believe my brother has all three so I could easily get hold of them. Dunno if they're HD remakes though. 

Do you have to play the others first?  No...but you absolutely should if you care about the story at all.  God of War has the best story of the trilogy in my opinion and is a must play. 
 
Edit:  Also, you should play them in order if you plan on playing all 3.  Parts of the story are given away in II and III if you haven't played the original.

You would definitely be a bit confused if you jumped right into GOW3, seeing as it picks up immediately where the previous game left off. That being said, the story in the GOW series is not it's most compelling feature; in the end, it's all about murdering the hell out of the Gods of Olympus and Titans in ridiculously savage ways. Like Vito said, the first two games are definitely worth playing and you can probably find them pretty cheap now. It's up to you.

Yes you definitely should. I got the collection before God of War 3 came out in order to get upto speed. Suprisingly though, God of War 2 turned out to be my favourite in the series after I had played all 3. I'd say you could go without playing the first one and just read/watch the story on the internet, but you really owe it to yourself to play the second one. Best game in the series and easily one of the best games I've ever played.

I would say it's a good idea to start from the beginning, plus as you play the first game and get to the third you will notice how much the games have evolved. It's pretty amazing.

I'd play the other 4 first personally...which is what I'm doing right now!

I played God of War 1 back in the day but wasn't that impressed. I thought it was way too frustrating as much as it was fun. For the heck of it though I bought Chains of Olympus...which I ended up liking so much, I ordered God of War Collection. I put Chains on hold while I played God of War 1. I still see why I was so unimpressed the first time. It has a LOT of places-even some stuff in the first level-that feels like it wasn't play tested. That it's just super frustrating. People point to these towers with rotating spikes towards the end of the game as being stupidly hard, but there's a dozen other places like that. So...it's annoying, and I don't want to play it again, but I still liked it decently, and like having the story.

Anyway finished GoW 1 last night, started up Chains of Olympus again, and played the last three hours of that today. And then started GoW 2, which I'm about 5 hours into now.

Both Chains of Olympus and 2 look and play WAAAAAAAAAY better than GoW 1, which looks primitive having come from Chains. I'm not done with 2 yet, but so far everything in Olympus and 2 just feels bad-ass to play. I've had to refer to a walkthrough for a couple of puzzles (which were obvious in retrospect), but there's nothing in either game that's as frustrating even as some stuff in the FIRST LEVEL of the first game.

Graphically Olympus is a lot better than 1, and 2 (in the Collection version) a lot better than that. 2 looks so much better than 1 in this collection, that I have to wonder if Santa Monica deliberately made HD art assets for it with the idea that they might get used at some point. The video is much higher bit rate, the textures far better, the polygon count looks a lot higher, etc. Ironically it's one of just a handful of games that can run at 1080p on this generation of consoles. One doesn't really benefit, but somehow (maybe better textures or something) 2 actually looks really great like this.

Chronologically, the series goes Chains of Olympus > 1 > Ghost of Sparta > 2 > 3, so I'm playing them sort of out of order, but Olympus is soooooo much better than 1 that I don't mind playing it like this.

I'm not sure at what point this series started working it's magic on me...because on one level it's just a giant over the top spectacle with an angry white dude tearing things apart. But somehow I started really getting in to the story, and really enjoying the gameplay. I mean the fact that I was excited to pick Olympus back up after finishing 1, and then excited to start playing 2 after finishing Olympus... I'm forcing myself to take a break after 2, as playing all 5 games straight through is just too crazy to contemplate LOL.

Another random thought, if you like games along this line, I'd highly recommend Wolverine (the current gen version, not the last-gen port) and The Force Unleashed (ditto) which besides being really fun to play has an awesome story that feels like a legitimate missing film between Ep. 3 and 4. (Apparently Force Unleashed 2 has a really weak story...which is weird, but I bought it anyway because I loved the first game so much.)
 
Aaaaaand I'm done rambling for a while.  I've gone kind of nuts from these games LOL

If you can get your hands on the HD remakes easily I'd say go for it, it's a fun little history lesson (In game development not greek history lol) and I personally found all the behind the scenes stuff really awesome to see.  

The story in GoW isn't that great in the first place, so you should probably just play the third.

I disagree...it's pretty compelling IMO.  Heck, God of War 2 won Marianne Krawczyka BAFTA for her writing.  Superficially it seems...superficial, but I don't know, something clicked after a while, and I can't  imagine not playing all 5 (although GoW 1 is the worst by far it looks like).