나는 목욕부터 할께요
➔나는 목욕부터 할게요. This sentence is quite natural, but doesn't fit the situation you gave. This can be used when you choose take a bath among several things to do. In short, this is related to selection.
These sentences meet the situation you gave.
Informal and polite speech: 제가 먼저 목욕할게요(씻을게요, 샤워할게요).
Peer speech: 내가 먼저 씻을게(목욕할게, 샤워할게).