I need to a better set of lenses for my Nikon D5300 camera?

I just have Nikon kit lens of 18-55. I do shoot at narrow apertures to get the better sharpness. But I would like to improve even further.

I was thinking at prime lens 35mm by nikon f1.77

or sigma 18-35mm or 17-70mm How is the sharpness from these?

I do not care about shooting videos much.

Also to increase the zoom range i was thinking about Nikon 55-200mm,

the old 55-200mm VR is £125.00 the new 55-200mm VR II is £250 is this new version really worth of extra money? Is there a difference in sharpness?

If you are not getting sharp photos with your 18-55 mm, there's something wrong, other than the lens.

How an image looks has a lot to do with how well it is focused, the lens aperture used (two stops down is usually the sharpest lens aperture) and by using a shutter speed that is fast enough to prevent any blur caused by camera movement. Just the tiniest amount of camera movement can make an image appear unsharp.

Most people buy a prime lens like the 50 mm because it is a medium telephoto lens on your camera and is excellent for shooting portraits

Spend some time learning how to hold your camera in a way which will produce the best support for the camera.

Look on page 18 of your user manual and copy the drawings.

If you are trying to use your camera like a mobile or P&S, holding it at arm's length and using the LCD to frame your shots, this could be the total reason for your "out of focus" looking images. This only gets worse when you get longer lenses.

And yes, the 55-200 mm VR (either one) are fine telephoto zooms.

I agree with fhotoace, your lens is not the problem. The 18-55mm kit lens is a high-quality lens and is perfectly capable of producing excellent photos. Whatever is causing your sharpness issue will not improve significantly with another lens. By all means get another lens to extend your camera's "reach" but not in a quest for a dramatic increase in sharpness.

BTW, have you played around with the "sharpness" setting in your camera's menu?

If you look at the tested lens reviews at photozone.de and at slrgear.com you will notice that trying to improve sharpness by stopping down has its limitations. As you get above about f8 diffraction takes an increasing bite into sharpness. It also make for slower shutter speeds. Vibration reduction can help a little, but it has it's limits. A good tripod can help more. Be sure to turn off VR when using a tripod.
The D5300's sensor is capable of capturing very fine detail, but it also displays lens deficiencies, focus errors, camera shake, and motion blur at high magnification when viewed at 100% resolution.