There are many different ways that you can build a telescope with cardboard tubes, mirrors, lenses and some basic supplies. The easiest project is a simple refracting telescope using two lenses and two tubes. It will give you a fairly clear, magnified image, and might inspire you to do more. If it works well, try the link at the bottom for a more ambitious design.
Find two cardboard tubes. Get tubes that fit together fairly snug, with one just slightly wider than the other.
Secure an objective lens and an eyepiece lens. The objective lens is the one that gathers light from whatever you are looking at. The bigger it is, the better an image you will get. The eyepiece lens focuses the image into your eye. It should be fairly small, and should have a very short focal length. Both lenses should be convex.
Figure out the focal length of the objective lens. Hold it above the sidewalk and move up and down until it makes a small, bright dot. Measure the distance from the sidewalk. Do the same thing with the eyepiece lens.
Calculate the length of your cardboard tubes. Add the focal length of both lenses together and add a couple inches. For example, if your objective lens has a focal length of 12 inches and your eyepiece has a focal length of 2 inches, you will want your tubes to add up to at least 16 inches long.
Cut the tubes so that their length adds up to the number you calculated in Step 4. The bigger tube should be about an inch longer than half the total length, and the smaller tube should be about an inch shorter. So, for a 16-inch telescope, the bigger tube should be 9 inches and the smaller should be 7 inches.
Make a cardboard ring with an outer circumference that is slightly larger than the width of the bigger tube and an inner circumference that is slightly smaller than the width of the objective lens. Make another cardboard ring for the smaller tube and the eyepiece.
Glue the objective lens into the middle of the bigger cardboard ring. Then, glue that ring onto the end of the big cardboard tube. Attach the eyepiece to the end of the smaller tube in the same manner. Slide the two tubes together and look through the eyepiece. Adjust the length until you see a clear image.
Cutting your tubes to the exact right length isn't important. The main thing is that you make the telescope long enough.
Don't use an eyepiece wider than your tube. It won't improve the image, and it will increase the cost of the lens and the focal length of the telescope. When you are checking the focal length of your lenses, make sure not to focus the beam on anything flammable.