Handlebody decomposition and intuition I am trying to get an intuition of how to approach handle body decompositions.
I understand that a Torus can be decomposed into a 1 0-handle, 2 1-handles, and a 2-handle.  The 0-handle is a hole you cut into the surface.  Then there is a 1 1-handle that loops around the surface in such a way to form a "watch."  Then there is a 1 handle that loops around the surface the long way around.
With a 2-Torus it gets a bit more complicated.  If we split the torus into a tube, four holed sphere, and a tube we get the following.  We get for each tube a 1-handle and a 0-handle.  When decomposing the sphere, I'm not sure how to approach it.  My instinct is just 2 1-handles and a 0-handle.  One 1-handle to through the north and south poles and one 1-handle to form the equator.
Do I have to make any special considerations for the sphere since it has four holes?
 A: First of all, the terminology "2-torus" is incorrect. The term $n$-torus refers to a torus of dimension $n$. What you mean, I think, is a genus-2 closed orientable surface. 
Now, consider a genus-2 surface that looks like O==O (the two holes on either side, with a tube connecting them). Cut it through the middle like so, O=|=O --> O= + =O. Now you have two punctured tori, or, as you explained above, two tori each missing exactly one 2-handle, or 0-handle, depending on which side you're looking from). Since you want to look from bottom to top, let's say the first one has decomposition 0-handle + 1-handle + 1-handle, and the second one has 1-handle + 1-handle + 2-handle. Put them back together and there's your decomposition, without having to worry about 4-punctured spheres. 
However, just for fun, let's try to come up with the handle decomposition of a 4-punctured sphere (by the way, this is sometimes called a "pair of monkey pants" -- the waist, two holes for the legs, one for the tail). A disk is a once-punctured sphere, so let's start with a 0-handle. Adding a single one-handle gives us, topologically, an annulus, or a two-punctured sphere. If you see the pattern, let's add two more one-handles to up the puncture number to 4, and we're done. A 0-handle + three 1-handles is a 4-punctured sphere. 
