My gym has a steam room. What illnesses could I catch from other people in there if they have it? I.E. If they have a cold, etc.

USUALLY RESPIRATORY. Ailments and fungal infections of the feet, which love humidity.
Risk of Sauna. The same risks of sharing a closet with one or more complete strangers. The sweating is not the issue, but the close confinement with others bratheing is. Then compounded with an environment that would allow a viruse to survive longer outside of the body.