Your old D80 had a focus motor built into the body that connects to a mechanical coupling on Nikkor AF lenses to move the focus elements in the lens. Your D5300 does not have a focus motor in the body.
To use autofocus your D5300 requires that you use Nikon AF-S or AF-I lenses or the equivalent third party lenses (such as Sigma's HSM or Tamron's USD and PZD series of lenses) with focus motors built into the lens. The camera communicates the focus information to the lens electronically. When using AF or equivalent lenses (such as those you've listed in the question) with a D5300 you will need to focus manually.
This frustrating dichotomy is due to the way Nikon changed horses in midstream with regard to auto focus lenses.
When AF technology began emerging in the late 1980s, Nikon attempted to create a system that would allow old F mount lenses all the way back to the late 1950s to remain usable as manually focused lenses on the new AF capable bodies. In addition to retaining the mechanical linkage between the camera and lens to control the aperture and associated metering, they also chose to place the focus motor in the camera where it drove the focus elements in the lens via a mechanical linkage, rather than place the focus motor in the lens.
Another major camera manufacturer chose to make a clean break and create a new lens mount system with an all electronic connection between the camera and lens and to place the focus motor in the lens. The new "Ultra-Sonic Motor" design Canon used on all but their low end lenses soon proved to be far superior in terms of focus speed and reliability when compared to the mechanical linkage that Nikon, Pentax, and others used. So in order to remain competitive, in the middle 1990s Nikon added electrical contacts to their F-mount system and began creating AF-I and AF-S lenses with motors inside them designed very similarly to Canon's ring type USM. Nikon continued to place AF motors in most of their bodies as well to drive the existing AI lenses. When used with the newer lenses the focus motor in the camera is turned off and the motor in the lens moves the focus elements. Only when a lens with AF capability but with no built-in motor is mounted does the AF motor in the camera body engage.
When digital SLR cameras came onto the scene Nikon eventually chose to retain the in-camera AF motor capability only for their higher tier bodies. Currently, the D7xxx series and all full frame bodies include an in-camera focus motor. The D3xxx and D5xxx series do not.