Hey there,
First up with Nikon mount lenses. Any Nikon made lens that is not an AF-s type will not autofocus on your camera. Any Nikon DX lens will be fine but there are a lot of older lenses that will not be.
To summarise, there are a few things I think would be good for you to do:
- buy a decent, sturdy tripod - this will help landscape, macro and shots of your paintings
- don't buy any lenses just yet. The 18-55mm is a decent lens for learning some of the fundamentals of photography with a DSLR
- read your camera's manual to understand all of the settings on it. You could arguably get away with P or Auto modes for most of your photography but for great shots of your paintings you really should be in M
- get out and take photos with your camera. Think about what you're trying to do and the result you get on the screen afterwards.
- read. There are a lot of textbooks out there on the topic of understanding exposure and how best to use light on a subject and in addition, a lot of the sharing or "talk" forums in here have stickies with excellent learning information. If there isn't a sticky there to answer your question, do a search for a similar thread or start a thread with your question.
More lengthy discussion below...
Paintings: I can't help you with best lenses to take photos of your paintings. Do I understand correctly that you are photographing your paintings and selling those photographs? Or are the photographs advertisement for the paintings? If the first, I'm not sure a DSLR is the best tool for the job, however it could certainly be made to do a very good job. Others may be able to help better. If using the D3300, even the 18-55mm kit lens should be able to get you decent images. Set aperture in the f/8 to f/11 range, the camera on a tripod with its lens looking perpendicular to the painting to prevent perspective distortion and a couple of diffused lights. Better, more expensive lenses may allow you to do a better job of course.
Things that we would need to know to help you better are the size of the paintings you a shooting and the environment that you're going to be shooting in (e.g. painting on the wall or on the floor, amount of distance you can get from the paintings)
Landscape lenses: well it depends on how you want to depict the landscape you are looking at. Some like the ultra-wide look, being able to get lots and lots of the scene in the picture, others will like much longer lenses. You can shoot landscape with any lens as long as it frames the scene the way you want it to. I use various lenses from an old ultra wide zoom (an equivalent for your D3300 might be the Sigma 10-20) all the way out to super telephoto. Most of my recent landscape scenery has been shot in the very wide to slightly wider than "normal" (e.g. between 18mm and 24mm on your D3300)
Macro: If you want true 1:1 there are a number of specific macro lenses in the 35mm to 200mm focal length ranges ranging from the inexpensive Nikon 40mm DX to the Nikon tilt shift Micro's and the Nikon 200 f/4 Macro. There are also a LOT of older Nikon F mount macro lenses which are for the most part very good-excellent. Nikon also built the 70-180 Micro zoom lens which to my knowledge is one of the few genuine macro zoom lenses.
All of the fixed focal length (prime) macro lenses will be optically very good-excellent. Nikon, Sigma, Tamron and Tokina all make macro lenses which will work fine on your camera.
The shorter lenses (like the Tokina 35mm will be lighter to use but to get high magnification will require you to get really, really close to your subject (almost to the point of being right up against the front of the lens), while longer allow some more distance between the lens and subject. The longer lenses though are harder to keep steady at really close focus (they're heavy and on top of that any movement of the camera is magnified more than the shorter lenses) so require lots of practice and/or a steady tripod.
I think a nice balance for an inexperienced user with the D3300 would be a 60mm or 85/90mm. I personally use a Sigma 150mm f/2.8 OS HSM macro lens because I like the longer working distance.
On the other hand if you just want to be able to focus closely to a subject when walking around, there are a number of lenses out there which have a short minimum focusing distance - petty much any zoom from Sigma, Tamron or Tokina with Macro in its name won't be a true macro lens but will allow you to focus pretty closely to a subject getting you interesting magnification.
Macrophotography by LordV is a excellent starting point for understanding macro and what you can use.
Zoom/distant/Zoo: There's so many options out there for this type of photography. Others may be able to help you better in terms of good quality for reasonable cost but I've heard good things about the Nikon 70-300 VR and the Tamron 70-300 VC lenses. They'll allow you to "zoom" in nicely on subjects at a distance and won't be super expensive. Telephoto shooting can be fun, but can also be challenging. Shooting moving subjects in darkish environments with the 70-300 lenses will be tough to get a nice image without lots of noise or subject blur. This is one of the areas of photography where getting lots of light to the sensor is only possible with lots of $$. At shorter focal lengths there are a lot of inexpensive lenses with wide apertures but there aren't any inexpensive telephoto/super telephoto lenses with wide apertures.
