When you take a warm shower and there's steam in the bathroom, is the steam water particles?

Yes. What is comonly called "steam" in this case is not steam, but is entirely water paticles. In fact, little if any actual steam comes out of a shower, if it did, you would be horribly burned. Let me break it down:

* Out comes hot water.
* The hot water heats up the air.
* More water can dissolve in warm air than cold, so some of the hot water is absorbed where the two meet.
* But warm air rises, and as it does, it hits cooler air, cools down, and has to drop some of the water it just absorbed.
* Water molecules clump together into tiny droplets. We call these "steam" because the same thing happens when we boil water, but this is not steam. Steam is gaseous water. Steam is invisible. "Tiny suspended water dropplets"  are visible, but that's not a very catchy name.
* Warm, rising air carries the droplets upward into drier air, so they stop growing and start evaporating again.

I say "little if any" actual steam comes fom the show head, because temperature is an average. If you water is 120 degrees F on average, some of the molecules are probably hot enough to be gaseous, but they will quickly cool and play along with the others. If you set the water heater anywhere close to boiling, and it didn't cause it to rupture or blow a pressure release valse, and you took a shower with that water, you would have a very bad day.