The other important rule to is to stay hidden in the shadows. Always conceal yourself and make sure that the circle in the lower-left hand corner of the HUD is dark, or you'll risk getting spotted by any number of unseen guards on high ground. You don't always need to stick to the shadows, especially in smaller areas with fewer guards, but for big hub sections you will want to neutralize all forms of light. You can do this with an action as simple as turning off a light switch and snuffing candles with your fingertips, or shooting a potent water arrow at an open flame. Water arrows should be conserved for more high-traffic areas, but you can switch off pretty much any light indoors. Don't underestimate the importance of dimming a candle in an empty room -- a guard may walk in at any moment and see you.
Mobility is a bit strange at first in Thief, because it's essentially a first-person Assassin's Creed. You'll use one button to run, climb, and initiate jumps. When looking for spots where you can climb vertically, try to locate areas with white chalk-marks on them -- they signify a safe zone to pull yourself up. If you've hit a dead end and have no idea where to go, look for a crack in the wall and hold X/Square until you find it; oftentimes Thief will make you go through a tiny tunnel to reach the next area.
In terms of picking up upgrades, get the wrench and the painting knife as soon as possible from the item vendor. They'll pay dividends with more gold, as you can use them to reach new areas or obtain new items that you otherwise couldn't before. Your Focus abilities are another matter, and your first inclination should be to upgrade the "Stealth" tree, which makes you less noisy all around and improves pretty much all of your skills at once. One level of Intuition and Dexterity are also a safe bet, since they'll inherently earn you more gold as you'll be able to pick locks and swoop in to grab items more easily. If you lose your way on the hub map and can't find the two upgrade vendors, remember they're on the Northeastern part of the screen.
Remember how I said entering combat was bad? Well, sometimes it's unavoidable, and you'll need to step in and take care of business. To reiterate, Garrett is terrible at fighting, and all he really has in terms of close combat is his blackjack club. If you find yourself entering combat far too often and failing, upgrade your Focus abilities to access the special "one-hit stun" power and power-up your blackjack's strength from the item vendor. You'll find that with a full Focus meter, you can take down at least two guards at once without losing any health.
In the challenge maps, which are a large part of the game separate from the story, stunning or fighting guards detracts from your total point value. It doesn't automatically boot you out of the challenge, but it will severely impact your score to the point where it's not even worth posting on the leaderboards. If you're playing the "Chain" game mode, which rewards you for stealing items quickly in succession, one of the best ways to earn a high score is to scout rooms before you start taking everything. That way, you'll know where every piece is so you can grab item after item in a short amount of time, maxing out your chain.
Thief can be as easy or as difficult as you want to make it, but even with that in mind, the easier settings can still be pretty challenging if you aren't up to snuff with the ins and outs of stealth, and the top-end custom settings are as hard as they come. With a little practice and a lot of sticking to the shadows, all the loot in the world is yours for the taking.