Physical tips: Increasing your stealth level
There are different levels of stealth, which explains why you can see level 20something characters in stealth clearly but can't see level 70s until they're right on top of you. According to WoWWiki, which is good at stuff like this, stealth and stealth detection are measured in invisible "subtlety" and "detection" stats. The Subtlety stat increases by 5 with each level for rogues and druids, so that's kind of your baseline. Subtlety decreases the range at which players and mobs can see you or "notice" you (when mobs turn towards you and freeze for a few seconds.)
Unfortunately, everyone has a competing stat called detection, which also increases at 5 points per level. This lets people see you at greater distances. That's why 70 warriors can detect 56 druids easily -- their detection skill greatly outranks the druid's stealth skill.
But there are number of things you can do to increase your stealth level. For starters, you can be a night elf. The Shadowmeld ability gives 5 extra points in subtlety as a passive, "invisible" bonus, so it's like having an extra level of stealth. You can also take the Master of Deception talent in the (surprise!) Subtlety tree, which gives you 3 points in subtlety for every talent point you spend. Sure, it's more inefficient than the druid's 5 points in subtlety per talent point in their Feral Instinct, but at least rogues don't have to suffer the indignity of hunters attempting to tame them.
Aside from that, there are a lot of items that increase your stealth level by a certain number. A full list can be found here, but notable (as in obtainable post-BC -- I doubt you're going to get a group to go to Naxx these days) items include:
Assuming again that you don't go into Naxx, rolled a Night Elf, and put five points in Master of Deception, you can stealth like a level 76 if you get the right gear -- and like a level 82 if you pop the sneaking potion.
But there are also a number of things that increase your opponent's detection level. The most familiar is the human racial Perception, which gives them a +50 to detection for 20 seconds. The hunter ability Track Hidden gives them +30, the felhunter's Paranoia gives a warlock +30 as well, and the rogue talent Heightened Senses gives us +3 for every talent point. There are also a number of items that increase detection. (Before you yell at me for using Thottbot instead of Wowhead, finding items with certain auras is the one area where I still think Thottbot is easier to use.)
As you can see, there are a lot more physical tools to detect stealth than there are to increase it. So how do we still stay hidden? We use sneakiness to make sure that they don't even know we're there.
Sneakiness tips: Moving around in stealth
The easiest way to not be detected is to stay behind someone! Being in a 180 degree arc in front of the target gives you a much greater chance to be detected. Stay behind them.
You can't really do too much about the "stealth sound", aside from hoping that your enemy doesn't have their headphones on or speakers turned up.
In PVE, monsters will turn towards you or speak when they're on the edge of detecting your stealth. If you freeze and don't move at this point, they will usually turn away after a few seconds, allowing you to continue on. In PVP, this obviously doesn't work.
In PVE, you can distract mobs to turn away from you so you can sneak around the back of them, avoiding that 180 degree arc. In PVP, you can distract, but it's not quite as useful because it instantly alerts your target that there's a rogue there. Its best use is if they've already spotted you and are running away. You can distract them to turn their screen around and stall them while you catch up.
A lot of players have mods these days that tell them when their opponent uses certain abilities, even if said opponent isn't appearing on your screen or in combat with you. This means that if you stealth somewhere near these characters, they'll get a message that says "Xxroguexx: *Stealth*", which is like a neon yellow "Use your AOEs now!" sign. WoWWiki suggests that you stealth farther away and then loop behind your opponent.
This may sound cheesy, but hiding in bushes, behind trees, and on the other side of walls can really help you in PVP, whether you're trying to attack someone or just hoping that the higher-level players doesn't notice you so you can keep questing. You'll still appear semi-transparent if the player comes close enough, but because you're over a dynamic background, they might just pass you by. Unfortunately, there are no camo uniforms in WoW, so that won't help, nor can you paint your face the color of the bushes.
Do you have any tips on how not to be seen?