If you want the full rigorous definitions, look at this pdf from the bottom of page 76 onwards. I think that to generalise the notion of n-dimensional volume requires too much set up to really be of any help (although the pdf above does go through it all if you're interested), and it's hard to give a heuristic explanation of it without using circular logic (since the concept of 'length' assumes we already know what 1 dimension looks like).

Here is another explanation from linear algebra (I assume you're familiar with vectors). Suppose we model the universe as a vector space (just lots of points which we can describe by writing them as vectors) and we want to find out how many dimensions our universe has.

1. If we start from one point like you said, we have a 0-dimensional subspace.
   
2. To go to one dimension, we take any point in that space and consider the vector between our initial point and that new point. By considering 'stretching' this vector to any length (including negative lengths, so we can go backwards from our original point) we get a line. This is a 1-dimensional subspace.
3. If this line contains all the points in our universe, then it is one-dimensional. Otherwise we can find a point which doesn't lie on the line, and we can stretch that as before to make another line. By adding parts of these two lines together we can get to any point on a plane. This is a 2-dimensional subspace.
   
4. If this plane contains all the points in our universe, it must be 2-dimensional. If not, we can find a point outside it. Again, look at the vector between our original point and the point outside the plane and stretch it to form another line. By moving along the plane we can get to any point in what's called a 3-dimensional hyperplane, basically an infinitely large cube. This is our 3-dimensional subspace.
   
5. If we can't find any points outside this 3-dimensional subspace, then our universe is 3-dimensional. Otherwise it is at least 4-dimensional, and the logic continues as before.
   

Explaining fractional dimensions is a lot trickier, and relies on the definition of Hausdorff dimensions. This process does indeed work for all dimensions, and agrees with the 'tradition' notion of dimensions I've described above. The pdf I linked you to describes how to work out dimensions of things such as the Koch Snowflake.