Say if I had one GPIO pin. Could I connect it to another pin on a different Pi and have it pick up the first pin's signal as input? Would I need to connect both Pi's ground pins to each other?
It can be done several ways. You can connect GPIO pins as simple IO, or use the pins with UART capabilities for a serial connection (essentially like using a null modem), or use the pins with I2C capabilities for a master-slave serial bus arrangement. I'm sure there are other methods as well.
For a simple GPIO example, you could cable the 2 RPis directly together as you've described, with a pin designated as output on one RPi fed to another pin designated as input on the other RPi. The problem is that this is only unidirectional, so only good for letting one RPi know of a condition, and not any sort of response. Any given pin can be configured as either input or output, and won't have any way of automatically switching between modes without some extra work. If you want bi-directional communications, you'll need a second set of pins for traffic going the other way.
Always be aware that the RPi GPIO pins operate at 3.3v. Be careful if introducing other devices that might work at higher voltages. Also be cautious if using long copper cable runs if the RPis are on different grounds. At any sort of distance, a direct copper cable may not be your best option. You might want to use isolators or some circuit protection if doing anything complex. If you need distance between the RPis, network communications might be simpler.
tl; dr: Yes, it can be done. It may not be simple, but you need to expand on what you're trying to do to say for sure.
Yes, you can connect a Pi GPIO to another Pi's GPIO.
As you say you do also need to connect the grounds (so they both agree about the levels).
One GPIO should be set as an output, the other should be set as an input.
(If both are incorrectly set as outputs with one high and one low that becomes a possibly damaging short circuit).