Haven't climbed it myself, but I've thought about it and may go there some day. It certainly helps that there are cheap flights to Malaga.
I had a look at the
Mulhacén page. The information provided looks good enough to get started when organizing your own a trip. Among other things, it says you can hike up the mountain, which implies usual hiking gear is the minimum that you need. Route descriptions would be nice to have, but assuming that you can read a map, I reckon any decent topographic map of the area will be good enough.
Still, it's a big mountain. That means I would pack a little bit more than for a walk in, say, the Lake District. Even when planning on using the refuges, I would pack a fleece or a light down jacket to keep me warm if I somehow end up outside during the night; perhaps even a bivvy bag. An alternative would be to pack for a proper multi day trip and bring camping gear. That way a sudden onset of bad weather won't be a problem.