You haven't finished MGS3?? That's like, everyone's favorite and some say you'd go to hell for choosing any other MGS game.
MGS3's stealth is much slower paced than any other entry in the series. I remember the first time playing it in 2004 and had to virtually re-learn everything because it was much harder than MGS2, the initial learning curve at least.
You'll need to crawl most of the time and try to keep your camo index at 75-80%. Any lower and the guards will start to investigate. Having a camo index at 80% virtually guarantees you'll be undetected if you'e at least 5 feet away from enemies. Use the D-Pad when nearby enemies to move quietly, as moving the analog stick WILL produce sounds that will alert sentries at close range.
Also, keep you Stamina as high as you can. Forcing Snake to eat the same food will make him grow to like it, adding more stamina gain. If you don't heal when suffering from serious injury, your health won't reach 100% BUT you will heal naturally overtime and it will increase your life bar.
Finally, try to avoid using the free camera as I noticed that it makes newcomers even more disoriented. Try sticking to the original birdseye view, and use the right stick to slightly angle it to your advantage. And if you're forced into first person view because of tall grass, hold all shoulder buttons to peek up, very useful to scope out threats.
You'd be surprised, but MGS3 and MGS4 are the easiest of all the games once you get the hang of them. MGS1 and 2 on E.Extreme are a genuine nightmare.
Those are all the basic stealth tips I could think of, hope it helps!