I bought a Nikon D5200 and I'm willing to go out of the kit lens. So I stumbled upon two lens that I can afford, but I don't know which one to pick because I'm still a beginner. Can you please tell me the difference between these two lenses?
75% of my photos are portraits, so does one have an advantage over the other? Also, is the 300mm good for shooting the moon?
The two lenses you're looking at are two different types of lenses.
The 18-105 is a walkaround zoom, and can also be considered a superzoom (i.e., a lens with an extraordinarily large zoom range, that goes from wide angle to telephoto). It's made to be very convenient in terms of framing/zooming capability, but will have compromises in optical performance to cover the very large zoom range, and is liable to be "slow" (i.e., have a small maximum aperture). These types of lenses are typically good for travel, street shooting, and general walkaround usage in daylight, but may be less useful in low light situations, and may be limited in the type of background blur you can achieve with them. But it can be an all-in-one solution to covering wide-to-telephoto without needing two or three lenses.
The 55-300 is a telephoto zoom. This is the type of lens most people grab to get "more reach", or to get closer to faraway subjects. You can shoot the moon with it, although a supertelephoto (something longer than 300mm) would probably give higher image quality for those types of shots. Telephoto zooms are most commonly used for subjects like sports or wildlife, but can also be used for portrait shooting. However, lower-cost telephoto zooms, are typically going to be slower lenses with smaller maximum apertures. You would most typically use this together with an 18-55 kit lens to cover all the focal lengths from wide to telephoto.
The type of lens a lot of people consider a portrait lens for a beginner would be what's called a fast prime. A prime lens is one with a fixed focal length--it has no zoom capability whatsoever. It will typically have a much simpler optical design, but it will also be smaller, lighter, and probably cheaper than a zoom counterpart. And can have a large maximum aperture for less cost than an f/2.8 zoom would. Typical examples of this type of lens for a Nikon DX shooter would include the 35/1.8, 50/1.8, and 85/1.8 lenses. While some more experienced shooters can use or prefer to shoot only primes, these lenses are more often, for a beginner, supplements to their more versatile (but slower) zoom lenses.
My advice is for you to definitely go for the Nikon 55-300mm lens. Since you already have your kit lens 18-55mm , you can already take pictures of moderately wide angle , and natural human eye POV angle (which is around 35mm in a DX crop sensor).
Buying a 18-105mm will only extend your KIT lens's capability bit more by adding 50mm range. Buy this lens if you mostly do indoor photography. So you don't have to change the lenses when switching from wide angle photos to portraits. And you can get rid of your 18-55mm lens if you decide to buy 18-105mm.
However 55-300mm is an excellent portrait lens than the 18-105mm. It generates smooth blurred backgrounds for portraits in 150-300mm range. And yes you can use it to shoot Moon, and an excellent for wildlife photography as well (because 300mm in DX - which is the category your camera belongs to, is 450mm equivalent in a full frame). If you buying this, keep your kit lens because you need it for wide angle shots.
Since you already have your kit lens my recommendation is to buy a 55-300mm. For a casual photographer, that kit lens and a 55-300mm is all you need.
The higher the mm on a lens, the more zoomed in the picture. And 55-300 is a good lens, on top of the kit 18-55. 18mm can take a normal picture, it basically has no zoom at all. The 300 can zoom in a lot, and can take great pictures, but only close ups and far distances.
I'd suggest keeping your 18-55, and buying the 55-300.
As other said, 55-300 is a telephoto. It is a great focal range for outdoor portraits (e.g. you're trekking with a group of friends, and want to shoot portraits without asking: do don't need to get close to people). Note that Nikon's 55-300 is also a cheap, hence a low-end telephoto. It takes good picture, but the optical resolution doesn't give you much freedom for cropping your pictures after shooting.
The 18-105 already gives you a reasonable telephoto on the 105 end. For indoor portrait, it's already a relatively long focal. The optical quality of the 18-105 is much better, so you can cheat a bit by cropping the image afterwards to simulate a 150 mm or a bit more if needed.
See benchmarks on http://www.dxomark.com/ for more details about quality.
Nikon's focal length simulator can also help you to get an idea of what you'll get with each range of focal length.
Now, if you really want a great portrait lens, a zoom is indeed not the best choice.
