I noticed a bubble in our downstairs ceiling and there was water dripping onto the floor. I cut a small hole in the drywall to see if more would rush out but it seems to be just wet. There is insulation so I cannot tell how much there might be. The leak is right under our upstairs bathroom, and I had just finished a shower when I noticed the leak downstairs. Where do I start on diagnosing and fixing this problem?
First I would check thoroughly to make sure that you don't have water coming out of the shower onto a wall or floor. Maybe if there's a tub in the shower, water is collecting on the lip of the tub against a spot that has a caulk failure and is entering the wall. Look everywhere that's wet for problems.
If I couldn't find anything that way, I would try to remove the escutcheon in the shower to expose the valves and look for signs of water next to them. If you're lucky and you have an access panel behind the shower, start there.
If you have continued water and and you're relatively certain it's not just overflow or a tile failure, then you're going to have to open something up to find it. If it's a drain leak, you'll probably have to open up the ceiling.
I personally hate working on ceilings and would consider opening up a wall behind the shower before I opened up the ceiling (especially if it's in a closet and I won't have to do a perfect job matching paint)
As mentioned in comments, the other way to go here is just to realize you're already forced to repair the ceiling. Opening that up to a usable size isn't going to change the repair significantly. Since that's where the damage is, you should at least be able to see where the water is coming from and use that to plan the next move (or if you're lucky, repair from that spot) rather than just be guessing.