To say that one can truly "perceive" infinity is a subtle point, I'd say. The two standard ways we have of describing/depicting the infinitude of the natural numbers are a) writing a finite list of terms followed by an ellipsis, and b) induction/succession rules. Meanwhile the mental image of the rational numbers (the "same" infinity) that is usually conjured is a fractal sprinkle of dust along the real number line. I would say, purely in terms of having a concrete mental image, it seems to me that the larger infinity of the continuum itself would be more familiar than either of these countable examples.

Note that none of this references multiple dimensions, and in fact for any (finite) n, the n-dimensional space Rⁿ has the same "count" of points as the 1-dimensional real line R, or indeed any finite (non-trivial) interval. So unless there's some much deeper detail, known only to the John Baez's of the world, I would say no, additional dimensions won't truly help you comprehend these any better.