When we flush the bathroom downstairs, we can hear the toilet upstairs also flush. When the upstairs toilet is flushed, the shower drain gurgles (in the second floor bathroom). I know that something is wrong, just don't know what.
This is a classic venting problem. If it has NEVER worked right, someone designed something wrong and it's going to be expensive to fix now because it will entail opening up the walls.
If it WAS working fine and just started acting up, something is blocking your vent pipe. A common cause is a bird nest or dead animal (i.e. rat or squirrel) that crawled down into the pipe and got stuck. There should be a straight pipe coming out of your roof somewhere around the upper floor bathroom, that's your vent pipe. You can run a snake down that pipe to clear any obstruction or, if you don't like the idea of working with sewer pipes, call a plumber / rooter company.
Another possible cause if you live in very cold climates (and it is winter) is that snow has piled up so high that it has blocked the vent pipe. Or people taking long showers results in steam rising through the vent, then it freezes when it gets to the cold air outside. Over time that ice builds up and clogs the vent pipe.
If it is ice in the pipe, you can just pour hot water down the pipe to clear it. Then you have to take measures to prevent it from happening again.