I have a Nikon D3200 camera. When it takes pictures in high resolution, sizes can be up to 6016×4000 pixels. Is there any problem (apart from big files) having the pictures at this resolution? Would it pose a problem when viewing unless you zoom in? Should I resize them to a smaller size? If so, which ones are recommended?
Rule of thumb: you never have enough resolution.
You should always keep the original files that come out of the camera the way they are. If you need smaller or otherwise modified versions, create copies of the originals and apply those modifications to the copies. If you shoot RAW (which you should), this is usually the case, as the RAW files aren't image files anyway.
If you want to display the images n a website for example, you do not need that many pixels, because nobody has a screen this big. For that purpose, create smaller versions of your originals for web publishing. (this also reduces the file size that you mention, which is beneficial on the web, too)
If you want them to be smaller, you can lower the resolution. The setting is where you see "Raw+F" on this image, you could change yours to "Medium". http://www.gadgetguy.com.au/cms/wp-content/uploads/nikon-d3300-review-2014-06-08-450x230.jpg