Here's the final pictures of the painted mat that I made for my niece Alice's nursery. It came out wonderfully, brightly coloured and a bold addition to the already very colourful nursery. The only hitch, a missed heat register meant it was a little too big, so we had to remove the last two pieces for it to fit, but it can always be transported to a larger space later on.
I purchased the standard primary coloured foam matting at Canadian tire. I bought two packages of foam tiles that resulted in a 4'x8' mat. I primed the SMOOTH side of the foam matting - the rough side is treated and doesn't grip any paint. Two coats of primer did the job and then two coats of the base colour. For the graphics I used regular acryllic paints. Much like my other rug, I freehanded the painting, but you could always sketch it out beforehand. I used a random floral print with no repeating pattern to make things easier for me. I then sealed the painting with two coats of satin polyurethane which allows for spills and messes to be cleaned up. TIP: keep the tiles separated when painting on the primer, the base coat and the polyurethane, but keep the tiles attached when painting the overall graphic. This ensures the tiles won't be painted stuck and make it difficult to disassemble.
** Update: Since the feature this morning on Design*Sponge I've received a few inquiries about how the mat is holding up. The mat has been in the nursery for about a month now, and had been on my floor in the apartment for about 3 weeks prior to that. It's holding up perfectly. No chips and no cracking. With that said though, I fully anticipate that down the road there will be some small hairline cracks, due simply to the flexibility of the foam itself. In my experience with paints, I'm confident there will be no peeling or chipping of the paint, but the cracking is likely and I think will add to the charm and age of the mat. It will not alter the durability of it though since the substrate is good.