Usage of the rings of power are based on the experience/desires of the user as well as the limitations of the ring itself. In The Hobbit, when Bilbo first used the ring (in Gollum's lair) his greatest desire at the moment was to go unnoticed (i.e., invisibility). Frodo likely had the same experience since his initial understanding of the Ring was as limited as Bilbo's. Keep in mind that in the LOTR books, another character, Tom Bombadil also did not become invisible (or affected in any way) when wearing the ring likely due to his experience as a much older magical being. Thus, Sauron could use it to help create entire armies, whereas magically inexperienced Hobbits simply became invisible.
Also, what Oliver_C answered.