When I am shooting landscapes, where (at what point) should I focus when using Auto Focus?
Or if I am shooting the lit up skyline of the city, what should be my focus point?
The reason I ask is that in AF-S I have one small focus point, but if I am shooting the entire skyline, wouldn't it throw the rest out of focus? Or should I change to Manual Focus?
Normally when shooting landscapes it is best to mount the camera on a tripod for a variety of reasons. One of the benefits of doing so is that if your camera s capable of magnifying a Live View image you can focus manually on a central point and then spot check around the frame to check how well other elements in the composition are focused. Of course to get a true idea how focused other areas are you need to stop the lens down to the same aperture that will be used to take the photo.
Exactly were to focus, whether manually or using auto focus, depends on several variables.
Once you've answered these questions you can use a Depth of Field Calculator such as DoF Master, to compute the focus distance and aperture needed to include all of the elements you need to be in focus within the DoF. In general the best point to focus on for landscape photos is the hyperfocal distance.
Please note that the calculations are based on the assumption of an 8"x10" print viewed from a distance of 10" by a person with 20/20 vision. If you intend to create a 20"x16" print and it is viewed at the same 10" distance, the perceived DoF of the same digital file or negative will be narrower. The dropdown menu of various camera models also include a list of commonly used Circles of Confusion at the end. To determine what CoC you need to use for a given print size, see this answer. If you are using a cropped sensor camera, multiply the result of diagonal of the print in by the cameras conversion factor (1.5x for Nikon APs-C cameras, 1.6x for Canon APS-C cameras). For more about the relation of infinity focus, hyperfocal distance, and the CoC, see this answer.
You say "I have one small focus point, but if I am shooting the entire skyline, wouldn't it throw the rest out of focus? Or should I change to Manual Focus?"
This suggests to me that you have a basic misconception about how focus works, and that understanding that better will help with the whole problem.
No matter how it's done, a camera lens can only focus on one plane. Think of an imaginary, infinite glass wall parallel to the camera. The camera focuses at that distance, and by adjusting focus, you move that wall closer to you or further away.
When you use the red focus points — or the green square or whatever in live view — you are telling the camera's autofocus system to maximize sharpness (technically, contrast, but that's the easiest way to guess at sharpness) at that point. You find an element in the scene at the appropriate distance, and the camera automatically moves the "invisible wall" to the distance of that object. The key thing is that even cameras with 39 or 77 or 200+ focus points can't put the "wall" in more than one place. They just give you more options of places to select in the scene without changing your composition.
I said that only one plane — that invisible wall — can be in focus, and that's true, but given the realities of how images are captured and recorded, and just plain human eyesight, in reality, the wall has a certain thickness, where everything inside is in sharp focus. And it doesn't have hard edges — it trails off into more and more blur both in front and in back of the camera. This thick, non-hard-edged wall is depth of field.
Some point in that trailing-off, often after the physical limit of detection but before the blur is obvious, you as the artist might decide that the trade-off is acceptable for your use. Maybe there's nothing in the foreground to be blurry, so who cares? Or maybe you're just not expecting people to look very closely. So, your technical choices (aperture, sensor size, print size) together with your artistic choices determine how thick the wall is (and, correspondingly, how much of the scene is "in focus".
Lenses have a concept of infinity focus, which means that the lens is focused so that a theoretical object at an infinite distance would be in focus (not that everything is in focus — see What is "infinity focus"? for discussion). It works out that in this case, the "invisible DoF wall" is quite thick, so setting at infinity focus also includes a lot of the scene much closer than infinity. So, you could do that.
But if you think about it, if you're putting the wall infinitely far away, you're "wasting" some of the thickness of the wall — everything behind its center point is "even more" infinitely far away, and clearly useless. This is the concept of the hyperfocal distance — you can move the "wall" closer so that its back edge is at infinity, giving you more in-focus area in the foreground (see What is "Hyperfocal Distance"? for more).
A general rule of thumb holds that the depth of field "wall" extends about two times as much behind the exact plane as it does in front of it. That means that a decent guess is to focus about one third of the distance into the area you need to be in focus, and then stop down to get as much depth of field as possible. (Depending on the scene, you may not want to use the smallest aperture possible for the highest sharpness, but if even DoF is important, you still might — see Do smaller apertures provide more depth of field past the diffraction limit, even if peak sharpness suffers? for discussion.)
And, finally, for more details, in addition to the questions I've already linked in this answer, you may find the following questions relevant — recommended reading if you don't know the answer to each off the top of your head!
In general, if you have foreground elements, the best point to focus is about 1/3 of the way between the closest thing that you need to be sharp and the furthest thing you need to be sharp (or the hyperfocal distance).
I.e. the drop off in sharpness due to depth of field is asymmetrical and tends to drop off faster on the near side of the point of critical focus than on the far side.
In a landscape or a shot of a city skyline, most of the scene will be at "infinity" for the purpose of focusing your lens. With a 50 mm lens, for example, it doesn't matter whether a rooftop is 30 meters, 300 meters, or 3 km away, the focus is all the same. The same applies to a tree 30 meters away or a ridgeline 3 miles away. It's all optically at infinity.
The only issue is if you have forground elements in the scene, like using a overhanging branch of the tree you are standing under to frame a view accross cows grazing with hills in the background. The cows and hills will be at optical infinity, but the branch will need a different focus setting. Now you have some choices to make.
Since the point of a landscape shot is the distant objects, you may be able to just let the foreground be more blurry. It can even help focus attention on what you want. If that doesn't work for your scene, then you use a smaller aperture to get higher depth of field. That will make the foreground less blurry while still focusing at infinity. If you're really careful and know your lens, then you can focus a little closer than infinity and let the depth of field keep the distant objects in focus.
I've noticed newer lenses don't have markings for depth of field anymore, but old lenses would often show you what range would be in focus for different f-stops. Of course that was someone's arbitrary threshold for what acceptable blurriness is, but at least it was some guide after you did a few experiments with that lens and learned how it worked at different depths of field.
A tripod helps in keeping everything in focus since it allows the shutter speed to be slower and therefore the aperture smaller. If you don't have a tripod with you, and still want foreground and background fairly sharp, that might be a time to crank up ISO a bit and live with a bit more noise in return.
