Before you do anything else, go to this website and download a copy of the owner's manual: EOS T3i Owner's Manual
Read the manual and become familiar with your camera and its controls and menus.
Next, as previously recommended, do NOT use the "Green Square Full Automatic" Mode. Use "P" (Program) Mode instead to start out with. The main problem with the Fully Automatic mode is that the camera decides where to focus by focusing on the nearist object with sufficient contrast. This may or may not be the intended subject of your photo! Which leads me to my next tip:
Learn how to "Focus and Recompose". Read in the Owners Manual how to select a focus point. When starting out, it is most convenient to select the CENTER auto-focus point. Place the center AF point on your desired subject and press the shutter button half way down. This half-press locks in the focus for the photo. Keep the shutter button half pressed while you RECOMPOSE the photo to your desired framing. Usually, it is not a good idea to have your subject dead center in the frame, so recompose to get the best framing possible, then fully depress the shutter button to take the picture. Eventually, you should learn how to use all 9 AF points but for starting out, focus and recompose while using the center AF point is the easiest to learn.
I will add to this post as time permits, but for now, get the manual and start reading!
1.) Learning sources:
"Read the manual" is good advice, but I found that when I was getting started the manual didn't really help much. It is aimed at telling people who already know how to use a DSLR where to find the particular buttons on the particular camera. If you don't know what the buttons are or what they do, it is of limited value to know where they are.
You can buy a guide book for the T3i, such as "T3i for Dummies", or several others out there. That helps a LOT.
Or even better might be instructional videos. You can buy video guides to the T3i also. Something about watching someone do something is (for me anyway) much easier to comprehend than a written page. Google for it. You can also get a million FREE amateur tutorials on the T3i on Google Videos. Just enter a search like "Canon T3i tips tutorial" and you will see plenty of little 3 to 5 minute videos made by regular folks. Some are better than others, of course.
Just learn and understand the "exposure triangle". That simple diagram is the framework on which you will hang every other thing you ever learn about photography. Photography actually means "recording light" (basically) and the "triangle" is about the 3 things you can influence about how much light gets onto your sensor and what it looks like when it gets there.
Duration, Intensity, and Sensitivity. That's it.
Duration = how long the camera is open to allow in light = shutter speed (Tv).
Intensity = how wide open or narrowly open is the camera = aperature (Av).
Sensitivity = how sensitive to light (amplified) is your sensor setting = ISO
Everything is a compromise though. You generally cannot choose the best of all 3 in one shot. Have you ever seen a sign at someone's business that says "You can have it Fast, or you can have it done Right, or you can have it done at a low Price. PICK ANY 2." It is like that.
Learn how to shoot in M, TV & AV mode. Why get a camera like that and shoot in the generic modes? Here is a link to a great book that is written specifically for this camera. The guy goes through all of the options and recommended settings and even gives you things to do for practice.
Canon EOS Rebel T3i/600D: From Snapshots to Great Photos by Jeff Revell
Here's some info:
We filmed 2 videos using a Canon t3i with Magic Lantern firmware and were able to get some stunning results!
The first video was filmed in the souk of Beirut:
The second is a stop-motion shot at night with at least 4 seconds of open exposure and heavy laser play: