Higher dimensions are pretty hypothetical. Most of the time they appear in theory, they are involved in string theory (which I am far from an expert). My rough understanding, though, is the higher dimensions are looped.
For comparison, imagine that the universe was a hypersphere. Moving in one direction for long enough would bring you back to the beginning, a bit like how moving in one direction on a sphere will bring you back to the beginning after you have gone around that sphere.
Now instead of a hypersphere, imagine a hyperellipse, where the lengths of the axes are such that you could take a nice walk in a straight line, and end up where you began without having turned around.
My understanding is that the higher dimensions mentioned in string theories are so small that they loop around in incredible small distances.
A lot of smart physicists have been looking into string theories for years now, but those models are far from being as solid as quantum field theory or general relativity.
That being said, in order to move in another dimension, you would have to change your motion in that direction, and so somehow apply a force between you and an object that was already displaced in that extra dimension from our usual position.
Since all known forces act in the usual dimensions, I'm not sure how that could be accomplished.
TheAfter all that, the only thing that comes to mind is this: If there were two of the usual 3+1 space-times separated by a 5th dimension, maybe (and this is getting into scifi territory) a wormhole could connect them.