OP, your T3i and 5D are pretty similar... Both have nine points shown in the viewfinder and user selectable. Both cameras only have one of the more sensitive dual-axis type points... the AF point at the very center.
Now here's where there is some difference... The 5D has 6 more points that you can't see in the viewfinder. They are called "Assist" points and can optionally be turned on or off. They are clustered close around the center point... Holding the camera horizontally or in landscape mode, there is a row of three of these hidden points immediately above the center point, and a row of three immediately below. All of these points are within the Spot Metering circle, so they are close to the center point. There is a diagram showing them on page 76 of the manual.
On both 5D and 5DII, the two Assist points immediately above and below the center point are single axis/vertical type... when the camera is being held in landscape orientation. The other four are single axis/horizontal type.
The most important thing to remember about the Assist points, which brings the 5D/5DII's total number of AF points to 15, is that they only work in AI Servo focus mode. So they are mostly intended to help when trying to track moving subjects. They don't work in One Shot mode. Also, keep in mind that you have to enable the Assist points for them to work, via Custom Function III-7. Give them a try, if you are struggling to keep action shots in focus with your 5D. I don't use them much on my 5DII, but I simply don't use that camera very much for action/AI Servo photography. I have a pair of 7Ds that I use for sports/action/AI Servo. They are much better at it than the 5D/5DII. (The 5D/5DII are a bit better for low light and stationary subjects... and also for really big enlargements.)
If all your visible AF points are flashing when you look thru the viewfinder, you have the camera set to All Points/automatically selected. This might be called a "point n shoot" focus mode of the camera and leaves a lot to chance, that the camera will focus where you actually want it to focus. Generally the camera will focus on whatever is closest and covered by one of the AF points. That may or may not be where you want it focused.
Yes, a lot of use use the center point by default... But that doesn't mean you shouldn't use the single-axis peripheral points if it makes more sense to do so. If the subject is in reasonably good light, has decent contrast and some fine detail, the single-axis points can work just fine. Some of them are horizontal axis, some are vertical. Notice the dual-axis point in the center of your camera's viewfinder is indicated by a square box and the peripheral points by slightly smaller rectangles. The rectangle indicates the orientation of the single axis. A horizontal single axis will work best if focused on a sharp line that's roughly vertical (crossing it) and vice versa with the vertical single axis points.
The above example litle johny posted is excellent illustration of this... The center point would have focused on the people behind the subject. The far right point would have focused on the windshield of the parked car. The bottom AF point would have focusing upon the blurred objects right in front of the photographer, and all the people would have been heavily blurred. The AF point used for that shot is perfect, assuming the one person walking toward the photographer is the main subject.
In One Shot mode... used for stationary subjects... the AF point(s) actually flash twice. Once when the AF starts and a second time when focus is achieved and locked on. Often this happens so fast it only appears to be a single flash of red. The second flash of red is also "Focus Confirmation", letting you know that the camera thinks focus has been achieved. There is also a green LED that lights up in the viewfinder for confirmation. And, if you have it enabled, the camera will emit an audible "beep".
In One Shot, once focus is achieved, confirmed, you can recompose the image if you wish and the focus will not change so long as you maintain half-press on the shutter release button.
In One Shot, if you want to re-do focus for any reason (you moved, your subject moved, or you just want to focus somewhere else), you have to lift pressure off the shutter release button, then reapply it to cause the camera to refocus.
In AI Servo mode... used for moving subjects... the AF point(s) only flash once to indicate focus has started. Because AI Servo runs continuously tracking the subject as the distance from camera to subject changes, you have to maintain the active AF point right on the subject, focus never locks, and there is no Focus Confirmation. You cannot recompose because if you do, moving the AF point off the subject while focusing the camera will refocus on whatever is now under the AF point.
If trying to use All Points/auto selection with AI Servo, you are hoping the camera is successful passing focus on the subject off from one AF point to the next, if the subject is moving across the image area.
If using the 5D with Assist points enabled, it's sort of like having an extra large center point... in theory. That way if you have trouble keeping the center point on the subject, hopefully the Assist point will pick up on the subject.
There are three things that together make for good AF performance: the camera, the lens and the user. So far all we've talked about is the camera. Lenses also can make a difference... Canon USM lenses are faster and more accurate focusing than micro motor lenses. The newer STM focus drive lenses are somewhere in between. Sigma offers many HSM lenses, similar to Canon USM. And Tamron has begun offering USD lenses, also similar to USM.
Larger aperture lenses also tend to be faster and can make for higher precision (shallower depth of field). This isn't always the case, though. Some extremely large aperture lenses (85/1.2L II, for example) are deliberately designed to focus a little slower, in order to be more precise. This is called "long throw" focus, a term that dates back to manual focus lenses that were designed similarly, and you had to turn the focus ring a long way to move the focus much, but could do so very precisely.
Extreme lenses such as macro, many of which are able to focus all the way from infinity to full 1:1 magnification just a couple inches in front of the lens, also tend to be slower focusing simply because they have to move their focusing elements a long, long way. Conversely, very wide angle lenses that have a lot of depth of field hardly have to move their focusing elements at all to achieve focus, so might be very fast focusing even with simpler focus drives.
The third major factor effecting AF performance is the user themself. You have to practice and learn the AF system in your camera, how to set it up for different situations and how it will react in a wide variety of lighting conditions, etc. In general, keep it simple... I use single point the majority of the time, usually the center one... And about 80 or 90% of my shooting is sports/action with AI Servo mode.
There's also a technique used a lot by sports/action photographers called Back Button Focus. You might want to give it a try. It separates the focusing function from the shutter release, and puts the photographer more in control of AF. It's especially useful for action shooting and long telephoto lenses. But, once learned, really can be used effectively in practically any situation.
Not sure which of the online tutorials you have viewed... I highly recommend the AF system overview series of three that B&H has posted on Youtube, starting with this one. Watch all three! They are about 1/2 hour apiece.
Finally, Google "Hyperfocal distance" and read up on it. There's some explanation of this, too, in the video above. This is a technique where you don't actually focus right on the subject, but place the zone of focus to include the main subject and other secondary subjects, in combination with using various size kens apertures to control depth of field. Doing this well in the field is something you learn through practice... different focal lengths, distances to the subject and apertures render more or less DOF. Many lenses have a focusing scale on them that gives you a means of calculating a hyperfocal distance setting. Some cheaper lenses don't have this, and it's not as easily done with zooms. But you also can estimate it by eye.
Hope this helps.