You'll want the farthest jump range to minimize your travel time. Decreasing your ship's mass (all ships' hulls and modules have specified masses) and fitting a higher-rated Frameshift Drive (FSD) are the two ways to boost your jump range. Taking this into account, your best option for an exploration ship right out of the gate is a Hauler. It's one of the lightest and cheapest ships--almost an even swap from your Sidewinder.
Modules come in a number of classes and ratings. The classes are numbered and increase in size/mass as the numbers increase. Each internal slot in your ship is numbered, which denotes the maximum class size module that can be fitted there. The ratings are E through A, though they don't necessarily get "better" throughout the range--it depends on what you want to do. D-rated modules are the lightest in their class, so a well-outfitted exploration ship will be fitted with mostly D-rated modules. You'll want to keep your power plant an A-rated module (thankfully, these are also the second lightest in their class) because they run cooler and will help keep you from overheating during extended fuel scooping, or, heaven forbid, you jump into a close binary system and find yourself a bit too close to one of the stars. You'll also want an A-rated FSD as soon as possible, which will give you the greatest jump range. B-rated modules are the heaviest and aside from the FSD should be avoided entirely for exploration builds.
Coriolis.io is a great site for getting to grips with all the outfitting options, and that link will show you what you'd want for a basic exploration Hauler. The 1.8 million total cost is mostly on account of the advanced discovery scanner, which isn't a necessity, but highly recommended (it will show you all bodies in the system, whereas the basic/intermediate scanners will only reveal bodies within a specific range of your ship). The 200k detailed surface scanner is, however, a necessity, and it will pay for itself quite quickly, as it greatly increases your profits from exploration data sales. To use it, simply target a celestial body and fly close enough to it in supercruise for the scan to begin automatically. Scanning distances depend on size of the object, so stars will scan from much farther away than planets and moons. This type of scan is also how you get your name tagged on first discoveries, though you'll need to travel a good distance away from colonized space to start finding those (roughly 1k light years before you're practically guaranteed to find them). Systems can still be "unexplored" for you despite other commanders discovering them first, but no worries--you'll still be able to sell that exploration data for the normal price. You'll get a 50% bonus in credits for selling data for completely undiscovered objects, and it's only after you sell the data that your name will be tagged to them on the system map. Completely unexplored objects will be shown as such on the system map, but they'll have no "Discovered by" tag.
Don't bother scanning asteroid belts--they're not worth any credits and give you no first discovery bonus.
Fuel scoops are a necessity, since there are no stations out in the uncolonized void at which to refuel--you have to get all your juice from scoopable stars (only O, B, A, F, G, K, and M type stars are fuel-scoopable). The scoops have no weight, so they won't hinder your jump range, and their scooping rates increase per their letter rating per class, as do their prices. Be careful with this--a Class 3 D-rated fuel scoop will gather fuel faster than a Class 2 A-rated scoop, but the latter will be exponentially more expensive on account of its rating. Always shop for fuel scoops by their scooping rate rather than their letter rating.
You won't need weapons outside of the colonized "bubble" of space as it's very lonely out there, but shields may be helpful when you return, just for some defense against interdictions.
There are tons of other considerations, but these are the basics. Good luck out there! Go find a black hole. The Eskimo Nebula would be a good place to start. o7
There is also ~9 minutes of me flying through canyons with a large ringed gas giant on the horizon, if you'd like to watch that.
