I was brought Nikon D5200 18 - 55mm Kit Lens. Now can i upgrade to 70 - 300 mm Lens?

I was brought Nikon D5200 18 - 55mm Kit Lens. Now can i upgrade to 70 - 300 mm Lens? - 1

Yes, but be sure to get the more expensive one, the VR one.

Yes, a lens is just a lens
The mm is the focal length of the lens, simply put, it is the magnification of the lens. A longer lens will have more magnification than a shorter lens

Do you really need 300mm? A 300mm is rather long.

Try using Nikon's lens simulator, note that 35mm on FX and 28mm on DX is roughly what you see
Also note that any lens you put on the D5200 is going to be 1.5x longer than what is marked (google crop factor if you want to read more )

EDIT: Yes you will be skipping the 55-70mm range of zoom, think about it, you will have a 18-55mm lens and a 70-300mm lens, 55mm-70mm is not covered by either of these lenses
EDIT2: Well, technically 55.00001 and 69.9999mm are not covered

If you prefer not to have to carry two lens around, you could replace the 18-55mm with a 18-200mm or 18-300mm lens, which covers all the range of the two lens, plus more. The 300mm version is a bit larger and heavier, and more expensive.