mamateacher,
There is no easy answer for you.
As a T3i shooter, I'll do my best to help answer your questions.
I'll assume you are NOT using AUTO mode - AV (aperture priority mode) is a good setting for you to start learning with.
Shooting with DSRL requires understanding 2 main things: Exposure & White Balance.
Lets's Start with White Balance.
There is an "Auto" setting that will keep you out of trouble. Start with that so it's one less variable for you.
But it allows you to set for Daylight, Shade, Cloudy, Tungsten (i.e. house light bulbs), Fluorescent, and Flash
For daytime pics, you want to make sure that your camera's white balance is set properly... most likely set to Daylight or Cloudy. White Balance is what keeps the colors true in your photos - having the wrong white balance is what causes color casts in photos (example: photo looks all bluish).
Next is Exposure.
Exposure is all about getting light into your camera. There are only 2 ways to increase light into the camera - either enlarging the lens's opening (Aperture) or increasing the amount of time it is open (Shutter Speed). Think of the light like water... if the lens's opening is larger or left open longer, it will let more in. That's aperture and shutter speed. Aperture and Shutter speed must be decided upon not only for getting an exposure, but also for artistic considerations. Wider or Narrower apertures produce different results. Longer or Slower Shutter speeds produce different artistic results. Aperture and Shutter speed are like a see-saw... when one goes up, the other must go down but doing so produces different looking photos, all still properly exposed.
This is where the artistry of photography comes in. Look up Depth of Field (DOF) and Slow Shutter speeds to understand how to use these.
But with exposure, there is one more thing to consider - ISO... how sensitive your camera is to the light. Want to use a faster shutter speed to capture something fast moving but can't get a good exposure? Turn up the ISO. Of course, there is a trade off... increasing ISO increases the amount of noise (or grain) in your photo. Think of ISO like an amplifier... ever turn up the dial on a stereo even when no music is playing? You hear noise or "snow" in the speakers... that's how ISO works with pictures.
Can you recommend a setting for daytime pics with my family as well as nighttime?
Daytime shooting is easier because there is so much light.
Portrait type photos typically are taken with your lens at 50 - 70mm and an aperture around 5.0.
Landscape type photos need a smaller aperture for a larger depth of field. Push the aperture as small as you can to keep the shutter speed at or above 1/30.
Typically 1/30 is the slowest you can set the shutter and still hand hold the camera without any "camera shake".
As for your ISO, I wouldn't push the T3i ISO above 1600. You'll be there at night.
Also keep in mind that if you are using the kit lens of the T3i (18-55mm) it is not a fixed aperture lens... i.e. as you zoom with the lens, the aperture automatically gets smaller changing your exposure. At night, you may not have a choice but to keep the lens as zoomed out as possible to keep the aprture open as wide as possible to let in as much light as possible.
Also, I rarely use the camera's on-board flash. Believe it or not, the best time to use that flash is during bright, sunny days. The flash acts as a "Fill light" that helps reduce the harsh shadows on people's faces during high sun.
As mentioned above, check out the Photography board. There is a lot of talent over there but don't get discouraged when you see their amazing photos. You are only seeing their best... not the 100's they took and did not like AND most of them are pretty savvy with photo editing software too which helps take the images another step higher.
Finally... I bought my T3i because of the VIDEO capabilities as much or more than the photo capabilities. Learn to use your DSLR as a video camera - the results are amazing. My one tip for video would be to "hold still"... hold the camera still and let the action unfold in front of you. If you have to move, do so slightly and gradually.
Here is some Disney vacation stuff I've shot primarily with a T3i (there may be some point and shoot stuff mixed in):
http://tomryanmedia.com/index.php/videos/video/5.html
http://www.youtube.com/watch?v=YvtDFEtCHhU
Good Luck - Welcome to the Amazing world of DSLR!
It can seem overwhelming at first, but once you get the feel for it, you'll be glad you did.
