Years ago in math class I thought I heard that n dimensional objects live in n + 1 dimensional space.
So a point must at least exist on a line. A line must at least exist on a plane. A plane must at least exist in a three dimensional space.
Is this true? Is there a place I can learn more?
If by living you mean it could be embedded in an $n+1$ dimension, your answer is yes, we can extend the dimension and have our old object live in higher dimension.
But a line passing through the origin by itself is a one dimensional space and it does not have to be viewed as a subspace of a plane to be studied.
Same with a plane passing through the origin which is a two dimensional vector space and it is studied without the need to be seen as a subspace of a three dimensional vector space.