There are a lot of Nikon lenses that will FIT ON, and stop down to shooting aperture, and actually shoot photos on the D5300. But in order to have autofocus, your camera needs the AF-S or AF-S G model lenses, which are petty much the newer and newest lenses Nikon has made. Older, manual-focus Nikon lenses will mount and shoot, but you will have absolutely NO light metering, and no automatic flash metering, which is kind of a PITA, but those older lenses CAN be used, and there are a lot of oldies available through pawn shops, used camera outlets like KEH.com, and so on.
Using a manual focus lens on a D5300 would be a lot less convenient than a fully automated, AF-S G-series lens like the 35mm f/1.8 G-series "DX" model, which is under $200 brand new. THAT is a popular lens on the small Nikons, as a 50mm equivalent-type lens. The 50mm f/1.8 AF-S G is another good, under $200 lens that would be useful. Both these are light, small, and sharp. The 50mm has better optics--less color fringing than the 35/1.8 G DX. The 35mm G is the DX model; there is a brand-new 35mm f/18 AF-S G model that is **not**DX,but it costs more money.
The kit 18-55mm lens will get you started. It's an alright lens in decent light, or with flash.