Since the new mining system in Dublin is not the most intuitive and I've had a few people ask how to get started with it, I might as well document my answers so I can send it to them in a nice and low effort info dump. Hopefully all the basic questions get answered here.
First, all the regular mining rules still apply. You'll need a ship with a mining array, as well as the right ID. For how to outfit a mining ship, check the mining guide. You'll also need some actual weapons, more on that later.
Once your mining vessel of choice is ready, you'll need to follow these steps:
Go to the mining zone in Dublin (F6 on the map)
Switch to the Loot tab on your scanner, you'll see some Resource Rich Asteroids, go to the nearest one;
Shoot it with the mining array to "deshield" it;
Use your regular weapons to shoot off fragments of the asteroid to expose the ore nodes underneath (you can see which parts can be shot if you go to the subtargeting scanner window, they also have a slightly different texture with visible seams on the surface)
Shoot the gold nodes, they will drop ore, tractor it in with B;
Once the asteroid itself "dies", it will also drop ore. Hegemons get double the ore from this final drop;
Move on to the next rock until cargo is full.
A sped up demonstration of destroying an asteroid fragment to mine the nodes underneath, courtesy of @Tijno
But wait, there's more!
I mentioned that you will need some actual weapons for mining. For a Hegemon:
A pair of Tachyon lances is a decent bet - they are precise enough to hit both fragments and nodes and don't turbo annihilate the entire asteroid immediately, leaving you with mere scraps.
A Flak turret can also help due to its AOE damage by destroying several fragments at once, once you get used to its low range and high dispersion.
The Hegemon is quite difficult to navigate through the field, so the optimal strategy is to shoot all the nodes you can easily reach before 'killing' the main rock and moving on.
If you want to use a small miner, like a fighter or freighter, you will need to equip:
Regular hullbuster guns;
A pair of Cannonball missiles to use as mining charges.
Cannonballs deal increased damage to asteroid fragments and a couple of salvos from those will open up multiple fragments at once, due to their AOE damage. Then you will just have to shoot the nodes with the cannons of your choice until they drop ore. Be careful not to accidentally blow up the ore - Cannonballs can do that. You can team up with a Hegemon to speed up your big brother's efforts and use your superior maneuverability to get the nodes he can't see. Or you can work with a transport, leaving clusters of ore floating for your friend to pick up and haul away for selling.
Now you know the basics of how to be a proper space miner. We're rich!