MDF boards (you could use pieces of wood or old tiles)
Glass mosaic tiles
Ceramic mosaic tiles
With kids it's best not to cut the tiles, but use them whole instead to avoid any really sharp edges. I was lucky because we have a huge bag of ceramic tiles that have been in my dad's shed since the 70's - they're very retro... Also we used normal square glass tiles and some millifiori that I bought on ebay.
First step, get sticking those tiles on - random patterns work fine!
Then grout - you could colour the grout with acylic paint for a more colourful finish.
Give your mosaics a polish and use them as trivets or add a hanger to the back to hang them on the wall.