Hi and welcome!
As you might gather from the above responses, there are several possible reasons for blurry images...
1. Too slow shutter speed (camera shake).
2. Too slow shutter speed (subject movement).
3. Missed focus.
4. Too close, within the minimum focus distance of the lens.
5. Too shallow depth of field (too large an aperture for the situation).
6. Diffraction (too small an aperture for the situation).
7. Insufficient image sharpening.
8. Mirror slap.
9. Poor quality optics (lens and/or filter).
10. Damaged lens or camera.
11. Probably some other things I'm forgetting!
#9 and #10 are actually pretty unlikely and really are the only reasons for spending money getting gear repaired or buying new gear immediately. A poor quality filter, though, can simple be removed to solve any issues it might be causing.
#8 mirror slap (internal vibrations causing camera shake) only occur are certain shutter speeds, typically between 1/30 and 2 seconds. It's worse with longer telephoto lenses or high magnifications shots with macro lenses, usually mostly causes loss of fine detail that isn't very obvious unless viewing the image at higher magnifications. It's also not too likely to occur if you are using highly automated modes, because the camera is probably forcing you to use higher shutter speeds.
#7, all digital images require some sharpening. If using highly automated modes, the camera is probably applying reasonable amount of sharpening to JPEG images immediately. But sharpening is also dependent upon the size of the final print... if making a large one, more might be needed. Has to be done with care, though, since over-sharpening can cause odd looking images and add artifacts to an image.
#6 diffraction is an effect that occurs most obviously at really small lens apertures such as f16 and f22. It depends upon the density of the camera's sensor and also is related to final print size... the larger the print, the more obvious diffraction will be. With your camera, you are probably pretty "safe" from diffraction using f8 and f11, for most print sizes. It just goes to show, though, that there is a "Goldilocks" range of settings with your camera and lens, depending upon the particular situation being shot and the planned used of the images. Too small an aperture is not good, just as too large can be...
#5. Too shallow depth of field is caused by using too large an aperture for a particular situation (also by positioning your point of focus incorrectly... Google "hyperfocal focus distance" for more info). It's a little hard to be sure looking at small, Internet resolution pics, but your first two and fourth images all look to me like depth of field issues.... The main subject (i.e. the kid in the first two, mom & kid in the fourth!) appears fairly sharply in focus. The background in both images appear a bit blurred. Actually, that may be okay, since a slightly blurred background can help make a sharply focused subject "pop" to the viewer. It's a deliberate technique often used for portraiture, and one of the reasons many "portrait" lenses have quite large apertures available.
There may be some of this occurring with the third image, too... but it also appears the flowing water is further blurred by movement, suggesting a slow shutter speed (#2) was used. On close inspection, too slow shutter speed also can result in subject blur whenever photographing people, especially kids who are nearly perpetually in motion!
Yes, image stabilization (Canon calls it "IS", Sigma calls it "OS" and Tamron calls it "VC") can be helpful reducing #1... But is most valuable with focal lengths 100mm and longer. Stabilization can't help at all with subject movement, though... so you still need to be aware of your shutter speed.
#3 missed focus can be due to a number of things. Your camera has two different focus modes - One Shot for stationary subjects and AI Servo to track and maintain focus on moving subjects - which take some learnin' to use well. You also can choose between All Points/Automatic Selection and Single Point/Manual Selection of the auto focus point(s). If you use the automatic method, the camera will usually focus on whatever's closest and covered by an AF point (some cameras have face recognition, tho I don't know if the T1i is one of them). Using Single Point puts you more in control of where the camera and lens focus.
How fast and accurately a camera and lens focuses is decided by a number of things... the quality of the light when you are trying to take a shot, the method of focus drive the lens uses, the sensitivity of the particular AF point in the camera being some of the key considerations. Your lens uses a micro motor focus system, which probably fine for your purposes, but would likely be too slow for sports photography and probably too noisy for videography. Canon offers "USM" (ultrasonic motor) lenses for faster and most accurate focusing... and "STM" (stepper motor) lenses that are very quiet and smooth focusing for videography. Your T1i camera has nine AF points... the center one is a more sensitive "dual axis" or "cross" type. The other eight are single axis type. If having trouble getting the lens to focus, you might try restricting the camera to using only the center one.
Also, your 18-55mm lens is most likely a varifocal design... Most less expensive zoom lenses today are. This means that you always must refocus after zooming to make a change to the focal length. If using One Shot, the camera and lens achieve focus, and then stop and lock at that distance. If you change the focal length of a varifocal zoom without making the camera redo the focusing process, the point of focus will be wrong (the error might be hidden by depth of field if using a smaller f-stop).
I know I sort of skipped over some possibilities on the list above, but I think they are pretty obvious and don't need a lot of explanation... or may be beyond the scope of an online forum. The point is look at your own techniques to identify what's going wrong, what you want to see in your images, or if your expectations are even possible or practical. In time with practice and experience you'll learn how to manipulate your camera to get the best results possible, in any given situation.
I agree that a photography class... or perhaps some books on photography... might be a good idea since you appear to have an interest in doing better. Don't go buying new gear unnecessarily.... Likely it's more just a matter of you learning to do much better with what you've already got. Most cameras and lenses today are very capable... "upgrades" may only bring very incremental improvements and/or address more specialized needs.
One of the best things you can do is keep shooting! There's nothing as helpful as more practice and familiarity with your gear.