SpaceX's Landing Zone 1 is just a few miles down the road from their launch sites at Cape Canaveral, so they use a crane to put it on a truck horizontally and drive it back to a hangar near the launch site (at LC-39A) for cleanup, inspection, and refurbishment. Eventually, presumably, they'll be able to prep a landed stage for reflight there and mate it with a new upper stage and payload; right now they launch out of LC-40 next door.
For barge landings, similarly, they'll bring it back to land, then drive it to the refurbishment or relaunch site by truck. (This is how they bring their rockets to the launch site to begin with.)
For safety reasons, the landing site consists of just large bare pads of concrete -- nothing to crash into. While SpaceX hit within a couple of meters of bullseye on their first LZ1 landing, that's no guarantee they can repeat it consistently. The launch sites have quite a bit of infrastructure installed, including lightning-rod towers surrounding the launch pad, so they aren't appropriate as landing sites.
We're a long way away from being able to simply land, refuel, add a new payload, and take off from the same location as we do with commercial aircraft; I don't think there's any expectation that Falcon rockets will be able to do this.