We had a storm, and a GFCI outlet breaker flipped (orange light on). I reset it by pushing the button, and the outlet started working again. But everything else on the circuit remains dead. Can a GFCI outlet "go bad", or is there something else I should be looking for?
These devices do go bad, but when that happens they often will not reset and the outlet itself will not have any power. However, there are a lot of different designs, so its possible the LOAD side of the GFCI has gone bad, I guess.
Another thing that can happen when a surge is on the line is that already loose or poor connections can melt or break.
Your next step is going to get a multimeter and check for voltages along the circuit. This will involve frequent trips to the breaker panel for safety. Turn off the power when you are disassembling any wiring or taking an outlet away from the wall.
First, pull out the GFCI outlet (leave all wires connected) and check the LINE and LOAD terminals with your meter. LINE should always be hot, even when the outlet is not reset. Once it is reset, the LOAD terminals should have voltage. If they do not, replace the GFCI device (label line and load wires before removing the old one).
If there is voltage, you will just have to move down the line to the next dead outlet. On any other outlets the damage is likely to be melting or a loose wire. You're looking for the outlet that has power coming in, but not going out. Really scrutinize any outlet that just has wires pushed into the back rather than using the side screws - those push in connections fail the most.
You could do a test without a multimeter as follows. Remove connection from the load terminals and connect them to the line side along with the existing wires to the line. (This would be only temporary.)
Check that you have power on the GFCI receptacle, then check for power in the following receptacles, i.e., can you run lights or other loads on the following receptacles? If so, then the problem is in the load side of the GFCI receptacle. If you do not have power, then the problem is in the following wiring. If the problem is in the following wiring, it can be in the hot or the neutral side. To determine which you'd best have a multimeter.