I had a GFCI outlet in my basement that shorted out. I replaced it with a new GFCI outlet but now the outlet that supplies it (also a GFCI) from the upstairs bathroom does not work nor do the lights in the bathroom. However the two downstream outlets from the basement GFCI do work. What did I do wrong?

It sounds like the outlets that you are calling downstream are actually the original source of power to the outlet you replaced.
You may have attached the wrong conductors to the line side of the gfi. Remove the gfi that you replaced, test (if you have the proper testers and can do so safely)the wires and attach the wires that have voltage to the line side of the gfi and ensure that the wires that do not have voltage (with the outlet removed enitirely) should be placed on the load side of the gfi.
There is also a possibility that some other part of the circuit was damaged when the gfi "shorted out. What caused that?
Also, gfi's come with instructions. Are you certain that you have read and followed the instructions and checked the troubleshooting suggestions in those instructions?

Are the 2 GFCI's connected line -line- or line-load? If line -load, you don't need a GFCI in the basement, it is protected from the first one.