The Asp is generally considered to be the best ship for exploration and on paper it pretty much is, although often overlooked is the fact that the Anaconda can also be fitted for exploration, which really makes it the best explorer, just a very very expensive one. The Diamondback Explorer comes a very close second to the Asp, if you don't have the extra 4 million for the Asp, then go for the DBX.
When all's said and done though, pretty much any ship that can fit a decent FSD can be turned into a reasonable explorer. The exceptional jump range of the Asp and the DBX is often what is touted as making them top explorer ships. However it has to be said that if you are only exploring within the main spiral arms and in around the galactic core, then having a high jump range capability is not really an issue as most systems will not be more than a few ly away from each other anyway. It's only as you approach the edges of the galaxy or the spiral arms where the systems start to thin out a bit that the range between them starts to increase.
Whether you need a massive jump range capability or not depends on your approach to how you want to go about exploring. If for instance your goal is to fly to the other side of the galaxy and you want to limit the amount of time to do it, then you will want the biggest jump range capable ship and FSD you can afford. If however your primary goal is to just get out there and discover cool stuff, then any ship that can jump about 12-14+ ly should be more than sufficient. Some cmdr's when exploring will set their routing to economic, so that their route from A-B covers as many systems as possible on the way. In which case, having a huge jump range capable drive becomes somewhat redundant. Other cmdr's will prefer their routing set on fastest, maybe their aim is to see as many nebulas as possible but they still want to discover stuff on the way. Jumping 25-30ly at a time, they will get to their goals quicker but will miss out on exploring all the systems in-between those long jumps. However, given how massively vast the galaxy is, that really isn't much of an issue either. When it boils down to it, all you really need to be able to do, is just keep jumping to the next nearest system, which in most cases is usually as little as 5-6ly jumps.
The only things I can think of that are must have's in exploration ships are an Advanced Discovery Scanner, Detailed surface scanner and a reasonable fuel capacity that can last a few jumps before needing a refuel. Oh yeah, add a fuel scoop to that list too.
So whether you spend a lot or a little, just about any ship can be used for exploration.
But good choices for exploration are:
Hauler
Adder
Cobra
Type 6
Diamondback Explorer
Asp
Python
Anaconda