How can I learn sneaking effectively?
Solution 1:
Perhaps this is common knowledge, but it certainly tripped me up on my first foray into thieving.
By default, your character is running. All the time. (This is separate from sprinting, which makes you move even faster and affects your FOV.) Unless you prevent it, you even run while sneaking, and this understandably hinders the effectiveness of your sneak.
On the PC, you can stop running by either holding Shift or pressing Caps Lock (by default). Ceasing to run will cause your character to slow to a crawl (or so it will feel) and dramatically improve your stealthiness.
Solution 2:
The very first Sneak perk provides a whopping 20% increase to your stealth effectiveness for only one point. While additional investment has diminishing returns for the mere dilettante, that first point is essential to sneak around effectively.
Solution 3:
Perform Stealth Kills, it seems to raise the xp bar the most. Best way to do that is obviously to go nuts in the environment or visit random dungeons. Additionally, the first few Thieves Guild Quests don't rely heavily on sneaking.
Solution 4:
In addition to the other answers, there is also three Guardian Stones you can find near to where you start the game. One of these is the Thief Stone and will let you skill up the Thief type skills including sneak 20% faster.
Also, being well rested will give you another 5% quicker skill gain, assuming you don't have a ... particular type of blood.
Solution 5:
Townspeople are not good Stealth targets. Even with my Sneak of 50, they tend to either not notice me at all or notice me completely. The thing is that they don't have an AI package that makes them search for you, which is what the half-open eye indicates. That's also the stage of detection where it's easiest to get Stealth increases.
To raise Stealth, you're going to have to change your play style for a bit. Go do a dungeon, but sneak the whole thing. Don't rush bandits, try to separate them, stab one, and then run away into the shadows. Repeat a few times to finish off a room of bandits entirely without ever being detected and your Stealth will have increased a couple of times. Now do that for a few dungeons (and always Stealthing while walking in the wilderness), and you'll have a decent Stealth in no time.
Do this in combination with the other Stealth-levelling bonuses (Thief stone, always being Well Rested or at least Rested), and it'll increase even faster.
The greatest recommendation for this way of levelling Stealth is that it will also level the other skills you use regularly, which keeps the Stealth increases from increasing the level of enemies disproportionately to your actual set of skills. By simply changing your play style slightly to include Stealth, you maintain character balance and can still take on the enemies you'll face.