Indoor is an adjective, indoors an adverb, and inside a noun, adjective, and preposition (the same holds for outdoor, outdoors, and outside).
While indoor stands comfortably in its grammatical place, indoors and inside (and their outdoor counterparts) prove more difficult to differentiate as nouns. Indoors refers specifically to the inside of buildings (literally within doors), but inside can refer to any sort of enclosed state: one can be inside a tent or cave or box or building, but one would only be indoors when inside a building. Though indoors may have its uses (specifically when referring to a generalized state of being inside), inside is more precise, more versatile, and far more common. Nonetheless, if you use both nouns, be careful to keep comparative expressions parallel: I’d rather be inside than outside is far superior to I’d rather be indoors than outside.
When using inside and outside in the transitive, prepositional sense (e.g., He is inside the car), do not pair the word with of. Prefer It’s inside the box to It’s inside of the box.
The set phrase the great outdoors refers specifically to the wilderness of the somewhat untamed and quickly-disappearing natural world, not merely a backyard or small park.