Now that's something I can talk about! I've been bounty hunting since my very first day in Elite, so it's time to guide the little ones in the same trail I blazed.
When you start the game, you will find yourself inside the cockpit of a sidewinder. It may not be an Anaconda, but if you use it right, you can bring down even the mightiest of pirate (edge)lords.
Weapons
The sidewinder only has 2 weapon slots, so there's few options to choose from. You could go for a double laser, which will take the shields down very quick but take longer to chew through the hull, with the advantage of not requiring ammo, so you can keep fighting until you either get tired or your hull gets too low for your liking. You could also go for a double multicannon, which will take much longer to get through the shields, but once they're down it will chew through the hull rather quickly, with the disadvantage of requiring ammo, so you'll have to abandon the fight earlier than you might expect. You could also get both a laser and a multicannon, which combines the advantages of both weapons but once the ammo runs out you will have only one laser to defend yourself with. You could also choose other weapons, like the plain cannons or the fragment cannons instead of the multicannons, for example, both of which do huge damage and I actually recommend them more than the multicannons. Or you could go fancy and try using more exotic weapons like railguns, plasma cannons, missiles, torpedoes, etc, but those will consume a TON of your ship's power, will heat up your ship super quickly and will be very expensive to reload. That and the fact that railguns, torpedoes, mines and plasma cannons are only offered in the fixed variant, which I really don't recommend to beginners. And that brings me to the variants of each weapon. Most weapons come in 3 variants: fixed, gimballed and turret. I highly recommend you use gimballed weapons as a beginner, as they can move on their own to give you an aim assist. Fixed weapons deal more damage than their gimballed counterparts, but are much harder to aim. Turreted weapons are not recommended at all for beginners since they offer the least amount of damage and are completely automatic. The only advantage of turrets is they can shoot behind you and aim at places you wouldn't be able to aim at with gimballed weapons. Besides the 3 variants of weapon mobility, there are also 3 different kinds of laser weapons; those being the pulse, burst and beam lasers. Pulse lasers drain less power from your ship but also deal less damage. As the name suggests, they fire in pulses. Burst lasers drain more power but also deal more damage and fire in bursts of 3. I think you can already anticipate what the beam lasers do. They do the most amount of damage while consuming the most amount of power from your ship, and they fire continuously in a beam.
Outfitting
You're definitely gonna need a shield, because you don't want your beautiful sidewinder to be melted in seconds. Normal shields are good, but bi-weave shields are better. They're a bit weaker than their normal counterparts, but they regenerate much faster. Prismatic shields are not only extremely slow to regenerate despite being extremely strong, but they're also locked behind powerplay, and if you're a beginner then you haven't done powerplay yet, so don't worry about prismatics. You're not even gonna need them against NPCs. Besides that, I recommend some hull and module reinforcements, and also some cargo racks because you never know when you might need them. Don't fuss about other modules and definitely don't buy different hulls (reinforced alloys, military alloys, etc), they're gonna cost you the same it cost you to buy your entire ship, sometimes even more!
Stations
You could definitely start bounty hunting where you are, but if you want certain modules and ships, you're gonna be hopping around the bubble to find them. That's why people recommend Ray Gateway at Diaguandri as a good station for bounty hunters, because they have most of the modules and ships you're gonna need for this career. I personally live in I Sola Prospect at Brestla, because it has all the modules and ships in the game, with the disadvantage that they cost more than normal.
Ships
As a beginner, once you've earned enough cash in your sidewinder, you can buy a new ship. These are a few of the ships you could buy:
Eagle: basically a glorified sidewinder with wings. Very fast, but packs about the same punch as a sidey. I keep one in my garage more as a decoration than an actual fighter, but, just like the sidey, it could become very deadly in the right hands.
Cobra MK3: the dearest of the older Elite players, the same ship CMDR Jameson piloted. It's very good at everything, not just combat, and is basically a bigger version of the sidewinder. Very good, I recommend.
Viper MK3: this is my dearest ship. If I could marry a spaceship it would be the Viper MK3. It looks fantastic, is very quick and packs a punch. It can even be outfitted to be the fastest ship in the game, at about 950 m/s if I remember correctly. Definitely give it a try.
Viper MK4: I've never actually bought a Viper MK4, so I can't tell you if it's good or not. I've seen people saying it's bad and others saying it's good, so give it a try if you can. You might be surprised. According to the in-game specifications, it has a stronger hull but is also heavier and slower than the MK3. Aesthetically it reminds me of the Rangers from Interstellar.
Adder: this is gonna sound weird, but the Adder was actually the first ship I bought in the game. It's surprisingly very good, and, just like the Cobra MK3, it's very good at everything, not just combat. This was my first bounty hunting ship (after the sidewinder) and I also recommend it. It's very fun and unique.
Diamondback Scout/Explorer: they're both very good for combat. The Explorer is better, but the Scout is also very good. The DBX (Diamondback Explorer) is my second favourite ship in the game, only behind the Viper MK3. I love it and I recommend it a lot.
Vulture: also very good at combat, despite being a small ship it has 2 class 3 weapon slots, so you'll be melting your enemies very quickly... while also melting your own ship in the process. It suffers from crippling power deficiency, and it will be rare for you to see your capacitors at full charge unless you're not in combat. It is awesome though, not only it looks very cool and has amazing visibility, it also sounds awesome. Mine is called Lamborghini, lol.
Actually doing the thing
Alright, no more stalling. Let's get to the meat of this. If you want to start bounty hunting, I recommend starting at navigation beacons. Yeah, the resource extraction sites are better, but the navigation beacons are much safer for beginners and you're also not at risk of smashing into a rock. Once you drop at the nav beacon, you gotta scan the ships in the area until you find one that has the red "wanted" tag. This step is basically a press "G" simulator. Once you have the perpetrator in your scanner, get closer (and I recommend being at the back of the criminal ship, so that it doesn't fire at you right away and has to turn around to hit you) and fire away. If your shields start getting low, boost away and then come back when your shields are sufficiently regenerated. Unless the guy is getting beat up by the cops, in which case you shouldn't go too far otherwise the game won't give you the bounty. I don't know why it does that but it is what it is. Anarchy systems don't have cops, so you should be extra careful when bounty hunting at those systems. The cops are your friends, they will help take down the edgelord and you should be super careful not to hit them. If you accidentally hit them while targeting the criminal, they will just send you a message telling you to "look where you're shooting" or something like that, but if you hit them while targeting them, then the whole system will turn on you and you'll become wanted yourself. The same applies if you hit them too many times even if you're not targeting them. Once you're happy with your bounties, when your hull is too damaged or when you've run out of ammo, you can just enter supercruise and go to the nearest station, where you will repair, resupply and collect your bounties. Have fun, never fly without a rebuy (unless you really have to) and make your own mistakes so you can learn from them! I intentionally left this text very vague because, ultimately, it's you who have to decide how you want to play the game, otherwise you won't be playing, you'll just be following instructions, and that's boring. Oh, and one more thing: dont die! If you die, you lose your bounty vouchers!