It would have to be and in this instance because the two parts are complete thoughts. You could also use a semicolon or a period, but those are not in your options.
A is incorrect (I went to work, Stacy went home) because it is a run on and the sentences do not connect.
B is correct (I went to work and Stacy went home) because and connects the two subjects.
C is incorrect (I went to work but Stacy went home) because in order for the sentence to be correct, you would also have to insert a comma after work (I went to work, but Stacy went home).
D is incorrect for the same reasons as C.