For some reason, I'm having trouble visualizing how to put a CW structure on a disk with 2 smaller disks removed. What I'd like to do is have three 0-cells, five 1-cells, and a single 2-cell. Three of the one cells would be glued to the three vertices to give three disjoint circles, and the other two 1-cells would be used to connect the three circles together. Then I can't see how to glue the 2-cell in, because when I visualize it, it always ends up covering one of the holes in the disk.
I realize my ability to visualize CW-complexes is not so good. For another example, it seems to me that if you take a wedge of two circles A and B, and glue a 2-cell on by $AB$, you should get a wedge of two disks, which would be contractible. But using cellular homology I see this has a nontrivial first homology group. I'd appreciate help sorting these two examples out to aid my intuition.