Like all the great expeditions of history, the first big part of the journey is steadfastly acquiring the funding. Expeditions are expensive and it's very much the case that spending one more day earning more money for better equipment can save you TWO days of travel time due to superior gear (not to mention two fewer days descent into space madness!
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DO NOT LEAVE until you have an A-rated FSD. On a cobra this is about 1.6 million credits IIRC.
Do not leave without an Advanced Discovery Scanner. This is about 1.5 million credits, but it will pay for itself many times over in cold hard credits when you return. You will be throwing away millions of credits if you do not have one.
Once you have this ADS, it makes sense to equip a Detail Surface Scanner. This is $250,000, which isn't much next to the price of the ADS (I just see it as "exploration scanners cost $1.75M and fill two slots) but again greatly increases your ability to make money.
Get the biggest-class fuel scoop in one of the higher ratings (ie B-4 or A-4 for the Cobra). Fuel scooping takes forever with a small scoop, sitting there going nowhere doing nothing is a waste of your time, and a bigger scoop means you fuel faster from further away, which means you'll be less tempted to get in closer to speed things up, so you'll be less likely to have an accident... alone in deep space where no-one can help you.
So you're itching to get going, and making money is a pain? You can make money and test your gear and hone your skills all at once: just get the scanners - however you can - and mount a mini-expedition. Just 500 or 1000 LY out. That way you can do without the top-end FSD and scoop (get the best you can afford but don't sweat it if that's not much) then along the way do a lot of surface scanning of any stars and planets close enough that you don't need to fly over to them, plus any planets further out that look sufficiently valuable to justify some flight time. When you come back, depending on how much scanning you did, you could have earned a couple million, just on that little several-hour jaunt.
There's your A-grade Drive and some more upgrades.
Plus, you'll be learning first-hand about what you need and what matters to you in exploring, so you'll be able to tailor your ship to meet your priorities.
For the big trip, get an auto-repair module or two. They don't weigh anything, and can help if you graze a star and take heat damage to components. They can't help with your hull or canopy which are the most likely things to kill you, but they can help with things that are still pretty important (like your FSD). They can't repair themselves, so having two means each can repair the other, if necessary.
Heat sink launchers can help prevent heat damage in the first place if you're calm enough to use them, but they have weight and limited ammo, so explorers decide for themselves.
Shields are very heavy and don't protect from any of the dangers of deep space, but they offer safety when returning to civilized space to sell your data. Most explorers equip light shields, many equip none. If you return to a far-flung outpost like 17 Draconis, I doubt you'll need shields against other ships, but weeks in the void will have taken a potentially severe toll on both your docking skills and your hull integrity; shields can save you from docking accidents.
Good luck!
I've done a lot of local exploring in the Cobra, and I'm currently acquiring funds to equip my Asp for a trip to the core. From the local trips I learned that for anything deeper I want better jump range than the Cobra can offer, so I'm going to make money doing other things for a while. Equipping an Asp is even more expensive than the Cobra!
