Autonomous guidance requires two things. First, you need sensors of some kind to tell you your flight parameters: position, direction, and speed. Second, you need some way to modify the flight. This is common to any kind of autopilot or military system.

Sensors include satellite guidance including GPS systems which tell you position, and by taking two position measurements at different times you can infer direction and speed. Some sensors don't require satellites at all like barometric or radar altimeters, inclinometers, which measure the direction to the ground, and "inertial measurements units" which include accelerometers and gyroscopes to measure changes in direction and speed. A military system which is designed to be highly reliable will probably include many different types of sensors while a commercial plane might not have as many.

Then ways to modify flight vary depending on the device. On a missile it will mainly be moving of fins, on a plane there are flaps, rudders, and the throttle, on a drone there are also flaps and throttle.

Algorithms are created so that based on the sensor inputs the system knows how it needs to change its speed or direction to reach a certain location. Then this is translated into commands that go into change the angle of fins, engine speed, and other things to change the speed and direction by the necessary amount.

This is an interesting and complex question that involves at least 2 aspects, (many more really, but you can learn about all of those when you become an aerospace engineer). First, let's talk about autopilots, because you were right to suspect that autonomous flight involves autopilots. Autopilots are a type of control system. A control system is anything that automatically regulates something else through signals and sensors. You're body is a great example of a very complicated control system. For example, if you are walking and you trip, your body will automatically move and rearrange itself in an attempt to stay upright, or when you put your hand on a hot surface you often pull it away before you even realize that the surface is hot. That's the brain using signals from your senses (sense of touch for example) to control your movements.

Engineers can do the same things with complex machines, like airplanes. There are plenty of sensors on airplanes that can measure things like the angle of the plane, its altitude, speed and a whole host of other things. What a control system does is take the signals from those sensors and then uses that to direct the plane in some way. A simple example might be a control system to keep the speed of a plane stable (similar to the cruise control in a car). There are sensors that can measure the airplane's speed. If that sensor tells the control system that the airplane speed is decreasing, the control system will then send a signal to the engines to work a little harder to get that speed back up. Similar systems (although more complex) can be built into airplanes for all the controls needed to fly. (FYI, commercial aircraft that people fly in are required to have 3 separate and different control systems for flying the plane, just in case one fails, the other two still know what to do and so errors can be found and ignored.)

So that is very briefly how control systems work (there is a whole field of science dedicated to building and improving them). Now onto GPS. Many control systems work with GPS as one of the signals they use. GPS is short for the global positioning system and it involves a series of satellites in space with very precise clocks in them. The satellites broadcast a signal telling everybody what their clock says. A GPS device works by picking up the signals from 3 or 4 of the closest satellites (depending on if you just want position, or altitude too). The GPS reads these signals, which are really just very precise times. But the key is that this signal takes time to get to the GPS device. The signals from satellites that are further away from the GPS device will arrive later than the signal from closer satellites. Think about how you judge the distance of lightening. You see the flash, then you count 1...2...3.. until you hear the thunder. The delay tells you how for the lightening is from you. GPS devices do this, very accurately, with 3 or 4 sources. This lets the GPS device pinpoint exactly where it is.

And so, we come back to your original question, how to self guided missiles and drones work. And the answer is that they use flight control systems that use sensors to figure out where the aircraft is, and then use control systems to modify things like engine speed and rudder direction to get the aircraft to where it needs to go. One of these sensors is usually a GPS device.