How can I speed up exploration?

Much like @Yuuki said in comments, scans complete faster the closer you are (can confirm this), and asteroid/resource sites don't give much for bonus.

A couple things to know about exploration

1. The basic discovery scanner has a small range of 500LS, intermediate has 1000, and advanced will spot everything within the system. source
2. The Surface Scanner will increase your exploration reward by a decent amount, as well as give more information.
3. The range at when you can even start scanning objects varies between their types
   • Stars: 1000LS
   • Large planets: around 200LS (depending on size)
   • Smaller planets (earthlike): around 50LS (again, depending on size)
   • moons, mining areas, and other small objects: under 10LS.

So how do you use this knowledge? When you jump into an unfamiliar system (the star you jump into will be undiscovered), fire off your scanners and then begin scanning in an outward direction all of the planets. moons and other objects aren't worth much money, so you can just keep them as "UNEXPLORED" and it's probably fine (you still get paid if they are just left as discovered)

If you have the surface scanner, be sure to fire that off on planets, especially if they look like they support life (life supporting planets gets a pretty boost in reward)