Hauler! The little mini-van that could.
Just finished a trip to Ghost of Jupiter, approx 1300LY from Sol. Will be returning to inhabited space now, stopping off at any interesting systems with earlikes or water worlds and whatnot.
You can get a Hauler to 25LY jump distance fairly cheaply. Mine is at around 23-24 because i didn't totally optimize for jump distance. Just kit yourself with the lightest of everything except power plant and FSD. Power planet just lightest which will accomodate your power requirements, and FSD the one that will get you furthest (note, not always the best rated, sometimes they are heavier and can drag down your range).
The small fuel tank in the hauler isn't too bad. I had a good fuel scoop, i could jump close to 100LY without refuling, although usually did it more often to avoid being stuck without fuel accidentally.
Other good ship for exploring is the Cobra. For a few million you can kit one out good for it. Depends on your budget.
Of course, if you can afford an Asp, then that is the best exploration ship currently in game.
If you are in a situation where money is a concern, then your priority should be getting the Advanced discover scanner and detailed surface scanner. If you can still afford the Asp with those, then thats cool, otherwise, go for a smaller ship.