Navigation in the Star Trek universe is both complicated and simplified. For the casual viewer, they spew off a few numbers and zoom, the ship is off on it's merry way.
One of the most important concepts of navigation in Star Trek is the heading or course of the ship. There are three numbers of a heading.
The first is a horizontal direction that is measured in degrees. This is shown in the diagram by the red circle. This number ranges from 0 degrees to 360 degrees. Both 0 and 360 refer to the same direction, straight ahead, but it's improper to give a number that is 360 or higher. There was a major goof in one of the ST:TNG episodes where a number higher than 360 was stated, which basically meant turn around in a complete circle and then some. As you can see from the diagram, this number rises as you turn to the right (starboard). To the exact right you hit 90 degrees. Precisely to the rear (aft) is 180 degrees. Exactly to the left (port) is 270 degrees. Precisely forward is 360 or 0 degrees. Of course the ship doesn't just make right turns, that's just how the numbers progress
The second component of a heading, the part after the "mark" is the vertical part of the heading. The valid range for a vertical heading is much more limited. You have the same possibile numbers but using numbers above 90, but less than 270 degrees are technically invalid because using those numbers would invert the value of the horizontal part of the heading and mirror itself. for example, a heading of 90 mark 110 would equate to a proper heading reading of 270 mark 70. A straight down heading would be 0 mark 90, while straight up would be 0 mark 270.
The third and final component of a heading is the speed which is expressed either in a fraction of impulse (c- the speed of light) or as a warp factor, or simply warp for shorthand.
It is highly unlikely that headings like this will be used outside of combat situations. The reason behind this is that humans aren't computers, and it's pretty unlikely that they would know their current position and heading relative to a target heading. The exception to this would be when a flight control officer is relaying a course to a shuttle or another ship, since they would have entered a destination into the helm and are able to read the information from their console. You are much more likely to hear something like "Set course for the 2nd moon of the 3rd planet of the Vega system." than to hear a captain rattling off a series of numbers like that.
Human vs Computer
We all saw the movie where either the directors or producers thought it would be "cool" to have Picard sit at the helm and have a joystick pop out and have him take "manual control" over the ship in a tight spot. This of course is utterly ridiculous, more "cool factor" than common sense. The computers are much more accurate and sensitive than a human could ever be.
Even when a heading is entered, at warp speeds it's not a straight line. While traveling at warp, things are coming at the ship much too fast for a human to be able to respond in time to veer out of the way, not to mention countering the effects of the gravity wells of stars and other things along the way. Without being able to rely on the computer systems, a ship could never go to warp. Even at sub-light speeds, a starship has such a variety of thrusters and engines that a human would have difficulty controlling them all reliably.
Leaving spacedock would be the only time where a human could be trusted to actually pilot a starship on their own, and it's more common to have the computer handle it.
Shuttles are a lot more simple than the starships, but still, they have fairly powerful navigational systems that allow for long distance travel when needed.
Space fighters are a special case, they are pretty simple machines, but they handle much differently than atmospheric fighters because there's no air in space for them to use airlons or rudders. They have more simple flight computers than a starship or even a shuttle, and aren't meant for long distance travel. They rely more on telemetry feeds from ships or stations than on internal navigational systems.