These DIY instructions are based on my mosaic-making expertise (such as it is). They're intended for home crafters who probably don't have access to workshop space, fancy powertools, etc.
Large shards or fragments of glass can be swept up using a small hand broom.
The best way to clear small shards or fragments is to wipe the surface off with several layers of damp paper towels.
I use cement backer, such as Hardie® Backer Board, for all of my coasters and most pieces smaller than about 8" x 8". Once you get the hang of it, cutting backer board is relatively easy at home. Below are instructions for the score and snap method.
Materials
Steps
Using a ruler or other straight implement as a guide, score where you want to cut using a scoring blade. To make accurate scores, use a decent amount of pressure and don't try to move the blade too fast. After the first score, you likely won't need to use a ruler to guide the blade, it should glide along the existing scores.
Continue to score until you've reached about a third of the way through the backing board. There is no need to score all the way, or even most of the way through the board.
Place the backing board on the edge of your work surface with the score facing up. With one hand, hold the board to the work surface. Quickly break off the desired piece by quickly snapping downward.
There's two types of grout you'll likely end up dealing with as a mosaic artist: sanded and unsanded grout. Both types can be purchased at your local home improvement store, since they're the same types of grout you'd use if you were tiling your bathroom or kitchen. In my experience, the decision of which to use comes down to the size of the gap left between the tiles and whether or not the mosaic will be exposed to the elements (e.g. water).
Generally, follow these rules-of-thumb to decide which to use:
For example, I now only use unsanded grout for coasters since that results in a piece that's more resilient to constant spills and wiping. For larger wall pieces, sanded usually works well. However you MUST pay attention to the spacing between the tiles. If you've placed tiles closer together than about 1/8", you'll need to use unsanded grout. Sanded grout will have difficulty filling in the spaces.