Many people make mosaics from cutting up existing tiles. But as a ceramic artist with a kiln, you have much more flexibility in making custom mosaic designs.
One important advantage of making your own mosaic pieces is that you can shape the clay before it is fired. So instead of being limited to shapes you can cut or break, you can make any shape you can draw. For example, instead of making a "fish" out of broken tiles, you can cut one large fish out of clay, then surround it by broken tiles. This gives a much more "custom" and interesting look. You can even put texture in the fish, paint it intricately with underglazes, or make it 3 dimensional. In addition, you can round the corners of the individual pieces to eliminate sharp edges.
When glazing (or underglazing), you can glaze a little down the sides of the pieces to elminate the possibility of seeing the white tile edges after grouting. This is especially important when using a grout color that is very different from the clay and glaze.
There are some mosaic tile artists who prefer to cut their pieces to shape with a band saw or ring saw. (Ring saws are made especially for cutting glass and ceramic.) You can make very exacting cuts without chipping the glaze, and since the piece is already fired you will know exactly the color that each piece will be.
Usually when making tiles, a lot of effort is put into making sure they are flat. We have several tips specifically on that topic! But when making your own mosaic pieces there is an advantage to not having a perfectly flat surface. That is that the warped pieces will reflect light at different angles. This gives the piece a more interesting look, sometimes like each piece is a slightly different color, and sometimes sparkly as the light reflects off certain angles. Of course if the warping is too bad your surface may be more uneven that you prefer. So another possibility is to impress a light texture into the clay. Just moving your finger along the wet, rolled out clay, pressing harder in some areas than others, will give slight curves on the surface of the finished pieces and create the same reflection effect.
One nice tip I learned is to put your rolled out slab of clay onto a piece of paper (something like butcher paper.) When you make the cuts, some clay will be pushed onto the paper and leave an outline of your design on the paper. This is very helpful for putting your mosaic back together! As you lift each piece off (in the leather hard stage), carve a number into the back side, then write the same number onto your piece of paper. Voila! Your mosaic will be easier to put back together than a puzzle.
Also, if you want to make many, many square pieces the same size, go to a home improvement stores and look for some plastic square lattice like covers for recessed florescent lights. They are White plastic grids with hundreds of little squares. Apply oil and press into clay and voila! Hundreds of perfectly shaped squares. In fact, if you don't want to deal with glazing all those little squares individually, you could probably push through only about 2/3 of the clay, leaving the bottom intact. Then you could glaze the tiles in larger sections and break them apart after firing.