Kaleidoscope Basics
Two things are important with a kaleidoscope. The mirror chamber and the image chamber or object.
The image end can be a marble, oil wand with floating objects, flowers pressed between discs, fused glass, a chamber with any number of small objects, the list is endless. You can even look through the mirror chamber at your surroundings and make it the image that reflects in the end. (Imagine reflecting beads, pop cans, trees, etc.)
The construction can be as simple as bathroom mirror inserted into a Pringles can, with a hole poked in one end and the other end used to hold an image, or as deluxe as fine grain wood, elegant stained glass, metal, polymer clay or other finish of your choosing. The shape can be a triangle duplicating your mirror configuration, a square or rectangle or a circle. The design is up to you.
The mirrors are what makes a kaleidoscope what it is. You can choose a 2 mirror or a 3 mirror kaleidoscope. If you make a 2 mirror scope it will need a piece of black cardboard, or glass spray painted black to provide the void on the third side. The resulting image will be a circle made of many triangles of your image. The remaining space will be the black void. (I've never tried making the black a color, but it might work as long as it is not reflective.) A 3 mirror scope will reflect your image repeatedly filling your view completely. Average length is 8-10 inches. If you want to make a scope longer or shorter you will need a lens on the end.
There are also mirror choices. If you just want to see a reflection and make something new, bathroom mirror is fine. If you want to produce works of art, I suggest using front surface mirror. Any glass store should be able to get it for you. If you have questions about the difference feel free to write me.
One last point that might be of interest. If you are very brave you might want to try a square mirror configuration or something other than a triangle. Some shapes are appealing, some not so much. Triangle is by far the norm. The shape of your mirrors will produce the shape of the reflected triangle. So, if you use a triangle with equal size sides (equilateral), the image will reflect with that shape of a triangle. If you like Christmas tree shaped triangles (isosceles) it will reflect that shape. If your triangle has a 30˚ angle on the end it will reflect 12 times around. (360/30=12 repeats in each circle at the end) The reason this is important is if you want to see a lot of little images make your top angle small. If you want to see a lot from the image repeated a few less times make the angle larger.
Mirror Chamber Basics:
Make sure your mirror is clean and lay one piece reflective side up on two pieces of tape.
(Bottom side
of mirrors)
Carefully set the edge of one piece of mirror on the outside edge of the first mirror.

Lift the tape to the mirror and set the two pieces flat on the work surface.

Carefully set the edge of your last piece of
mirror or non-reflective surface on the second mirror, again lift the tape and
set them flat.

If using front surface mirror remove protective film.

Now, carefully form all three into a triangle and secure tightly with tape.

Wrap with foam, or padding as needed to keep the mirrors from moving inside your kaleidoscope body. Check for fit. If you are going to be soldering the eye end, remove the mirrors now, move to the next step and replace them later. If you are going to be gluing your ends on leave them in place.

Finish the two ends. If you are constructing a stained glass body this means covering the eye end with glass and solder. If you cover it heavily with solid solder (picture below), carefully drill a hole for viewing. If you want a bigger hole, wrap the piece of clear glass with copper foil and add to the end of the triangle with traditional copper foil techniques. Solder to the body. Remove all flux at this point before inserting the mirrors. The image end can have a glass covering to protect the mirrors from dust and probing fingers or it can be left open. If you removed your mirrors to solder. Replace them now.

Enjoy.

Good luck.
Other body materials might have other options for end pieces. I have done kaleidoscopes as part of a classroom project that used pvc. By making the body out of pvc it allowed the students to glue a clear glass (or plastic) circle to the viewing end. Another choice is a connector piece with clear glass glued to the inside lip, pieces of glass, shell, etc to tumble in the chamber, and glass on the outside. The eye end can also be clear glass (or plastic) or an end cap of pvc with a hole drilled in it. There are many possibilities. Let your creativity soar.
~I will be improving this page as I get better pictures. :)