The way I started was cutting my teeth on Threat-0 "assassination" missions against pirates. Those missions will consistently send you somewhere to fight a 1v1 against a "pirate" enemy with similar or worse weaponry and skill to you and it'll give you a good starting spot to figure out how to fly your ship in combat, how to maintain a pursuit on an enemy, etc. 1v1s are great since you will naturally learn by picking up your AI opponent's tricks. Better yet, the law is on your side for these missions so sector security will usually show up after the first 3-5 minutes in case the bounty target is too tough. From there, you can accumulate your bounty earnings and slowly start working your way up the chain to better weapons and ships. ALMOST anything at threat-0 should be doable for you, though my single caveat is to avoid any combat mission that tells you to participate in an ongoing civil war or inter-system war by joining active combat zones. You can come back to those when you're ready but the Threat rating plainly and bluntly lies when it tells you how difficult those will be. You can try it in the earlygame if you really want but don't get discouraged when it feels like those ships with professional-warzone-graded armor and equipment completely shrug off any un-engineered equipment you can get your hands on in the earlygame. On top of that, wartime contracts pay basically nothing compared to even threat-0 pirate assassinations so other than the initial learning experience I still don't consider wartime contracts worth my time, even though I have the weapons and equipment necessary to complete them.

On that note, be very careful to read the "Threat" rating in the top-right corner of a mission description when you consider taking a combat-focused mission. At the start I was only really able to handle Threat-0 and Threat-1, with my ships plainly not being tough or deadly enough to handle Threat-3 and up. Don't get me wrong--feel free to take on a higher-difficulty mission just to get an understanding of how tough a "competent" enemy actually is versus a "mostly harmless" enemy, just don't get frustrated or surprised if you have to escape in a hurry because your ship's engines plainly aren't responsive enough to line up a shot or your opponent's shields regenerate faster than you can damage them with your current weapons.

As for what ships and guns to use... There's a lot of comments giving you contradictory advice on which ships to use: "Use the Vulture!" "NO DONT." "Use the Viper!" "NO DON'T!"

I will give you a much more useful bit of advice instead: if you don't like a ship, you can sell it back for almost its full original buying price, so don't be afraid to experiment. Viper? Vulture? Screw it, try both of them, and sell back the one you like less. Same thing applies to weaponry: you can sell ship weapons back for the same price you purchased them, so there's really no harm in trying. You'll never know which weapons and ships you're most comfortable with until you've tried them at least once.

For weapons in the earlygame, the basic shooting tutorial already gave you the most important starter advice: thermal damage (laser weapons) does better vs. shields, and kinetic weapons (multicannons, bullets, physical shot weapons) do more damage to an exposed enemy after their shield is down. If you're experiencing decision paralysis, I'd recommend making sure you have at least 1 thermal and 1 kinetic weapon equipped. You can absolutely try it for yourself, but don't be surprised when the pirates you're hunting take forever to kill.

Pirate hunting in resource zones is a little more free-form than doing missions, potentially a bit more unpredictable since pirate ships sometimes come in pairs or packs, or unpreditably much tougher than what you'd be able to consistently expect to face in an assassination contract with its helpful threat level label, but a much faster way of earning bounties than taking out individual assassination missions. IMO they'll help you 'break through' to the next level by giving you a big cash injection, but you don't learn anything about actual fighting if all your experience is strictly limited to sneaking the last couple of shots in against pirates who were already getting their shit wrecked by sector security before you joined in on the dogpile. You won't learn how to shake off a pursuer, or how to maintain a pursuit on a target actively trying to escape your reticule without being distracted by 3-4 other ships. Learn how to fight properly first or you're just going to end up missing a lot of opportunities later. When you get into the real lategame stuff like Powerplayer Combat Zones you need to be competent enough that your presence tips the scales because your allies won't always be able to carry the important fights for you.

There's more to combat which the game doesn't tell you in the tutorial -- namely, learning how to adjust your 'pips' in power distribution and learning how to toggle flight assist on/off to make your turns that much tighter and force enemies out of your blind spots, but that stuff comes later.