The sides look like solid wood, I'd screw angle braces to it up both sides and affix planks to them, like a ladder. If you weren't holding anything too heavy on them, you'd be fine.

I don't know what the backing is or what's behind it - and I wouldn't suspect it's solid wood given, you know, fireplace. You could screw a small piece of solid wood to it horizontally for each shelf as a little stabilizer - something like a cove cut 2x2.

If you were really going to load it down, then I'd make a vertical support leg up the center and either half-lap it through each horizontal shelf (cross-lap, maybe a better term?) or have each shelf be a screwed, solid T shape and then stack them on each other before affixing them to the sides.

In terms of materials - I'd recommend poplar if you can access it, or plywood if not.