I'm a little confused as to why it's 浴びる as opposed to する. In nouns like 掃除 or ゲーム, the verb used is する. Can anyone clarify?
The picture 浴びる should evoke in your head is "something pouring on to your body."
シャワー is literally water pouring on to you, so 浴びる is the most natural verb to go with it. Another example would be 日光(sunlight)を浴びる, and here you can also picture sunlight pouring down on you from the sky.
する is the most generic verb like "do", and as such it does cover a wide variety of cases. But again just like the English word "do", there's also a lot of cases where this verb just wouldn't do it. Examples include 医者にかかる, 車を運転する, 水を飲む, and so on.