I would like to determine the homotopy type of a torus with 8 punctures.
(I have come across this problem studying deformations of discontinuous groups of Heisenberg groups...)
Other than trying really hard to visualize, are there any other methods for finding homotopy types of punctured surfaces or the like?
A torus with one puncture is homotopy equivalent to the wedge of two circles, which is a $K(F_2, 1)$. To see this, think of the torus as a square with sides identified. Puncturing the square allows you to retract onto the $1$-skeleton.
It follows that a torus with $n+1$ punctures is homotopy equivalent to the wedge of $n+2$ circles, which is a $K(F_{n+2}, 1)$. To see this, start with the wedge of two circles and thicken some part of it back up to $2$ dimensions, then puncture it.