Nikkor AF-P 70-300mm DX VR would probably be the least expensive, at around $250 new ($150 refurbished). Confirm your camera is compatible with AF-P lenses, though. It's somewhat plasticky, but that helps it be relatively light weight.

Nikkor AF-S 18-300mm DX VR is a wider ranging zoom alternative, better built with better autofocus, for about $630 new.

Tamron 18-400mm is even longer, but not as sharp as some of the alternatives here, but better than one might expect from such an extreme range lens. It also has a lot of distortions... barrel at one extreme and pincushion at the other. It's the widest range zoom anyone makes right now and sells for about $650 new.

More expensive, but a lot more "reach" are the Sigma or Tamron 100-400mm. They both sell for about $700 new right now. Personally I'd choose the Tamron because it can optionally be fitted with a tripod mounting ring, which is an accessory I'd definitely want with a lens reaching 400mm (tripod ring sold separately for $129). The Sigma 100-400 doesn't offer a tripod ring option at all.

If these lenses seem "pricey", compare to the Canon 100-400mm or Nikkor 80-400mm lenses... or the Nikkor 300mm f/4 PF. Sure, these are all better than the above lenses in a number of ways, but they each also cost $2000 or more!

There is absolutely no problem with Sigma and Tamron lenses auto focusing on your D5300.

There is one Tokina lens I'm aware of that can't autofocus on a D5000 series or D3000 series cameras: the AT-X 100mm Macro).

There are some Nikkors too, that don't have in-lens focusing motors and would be manual focus only on D5300: "AF" lenses.... As opposed to Nikkor AF-S and (probably) AF-P, which can focus on your camera.