The first thing you will want to do is use your loan money to buy a Sparrow. You might think that a Star Barge would be better because you can carry more, but mining asteroids is much quicker with the dual lasers offered to you from a Sparrow.

You can do the starting quests for a bit of extra cash at the beginning if you want, but after that you will want to follow these steps.

-Buy a local map at New Boston in Ruticulus (unless you have a grasp of the area around the system)
-Go to the Ascella system (Two jumps right, one jump down from Ruticulus)

The main asteroid you're looking for looks like this:

Any other asteroids are also decent to mine. However, you'll want to avoid mining asteroids that look like this (These contain silicon, one of the cheapest mining materials in the game):

For new players, you can shortcut to land by pressing L. To sell your mining loot, you simply go to the trader on the planet and press the 'Sell all' button. You know you're getting the right materials if you are consistently picking up Uranium and Titanium, and avoiding picking up Silicon. If done right, you should be making about ~50K-75K per full cargo hold. Remember to pay extra on your loan every time you land. (Do this by clicking Bank>Pay Extra>Type amount) Otherwise interest stacks up and nearly doubles your initial borrow.

After you've payed off your debts and have any amount of money over 1 million, you'll be making your first ship change. Head to the Aldhibain system

and purchase a Fury at the shipyard. Remember to sell your Sparrow for a bit of the money back! Once you have your Fury, go to the outfitter and sell your four Energy Blasters and your Shield Generator. Remember that removing the shield generator does not remove your shields. You still depart planets with full shields, you just can't get them back until you land. An important outfit to put on any mining ship is an asteroid scanner. This outfit highlights nearby mineable asteroids and allows you to target them. Replace the guns with Beam Lasers and you're good to go! With the small amount of outfit space remaining, you can customize the Fury as you please. This ship might not seem like a large upgrade from the Sparrow, but with double the firepower, you can cut through asteroids very quickly! Return to mining in the Ascella system.

After you raise about 4 million credits, your next destination is the Alphard system.

This is a longer trip, so remember to make sure you have at least two hyperspace fuel before entering an uninhabited system. Don't be afraid to make an extra pit stop! Once landed in the Alphard system, go to the Shipyard and purchase a Hauler. Then go to the Outfitter and modify it as such:
Note: The Quantum Keystone isn't needed. I simply picked it up in alien space while doing some exploring.

You may need to search around for other outfitters for some of these modifications, but just know that your best bet for a good outfitter is the systems north of Alphard.

There isn't much to do but grind at this stage - the modified Hauler is a great method for mining quickly and efficiently. However, I recommend hiring a few extra crew and getting some laser rifles, because the Ascella system is frequently visited by weaker pirates, who tend to end up with their ship disabled. You can either sell these smaller ships for a quick cash boost, or keep them around to protect you and pick up some of the ores you mine (More on that in the Tips n' Tricks section). If you manage to get your hands at a Jump Drive (which I will not go into detail about how to do here, but there are many other guides that can teach you) then it is a great idea to move your mining operation to the Saquergen system, which contains the most valuable mineral in the game, Yottrite. This system is an isolated system found south of human space (see: bottom centre of image)

Once you reach the stage where you have upwards of 30-35 million credits in your account (made easy with the Hauler) you can head up to the Epsilon Leonis system.
Here you will purchase a Bactrian. (Note: This now requires you to acquire a city-ship license. It would take too much time to put a tutorial here but a tutorial can be found in other steam guides or elsewhere online. This may take a while, and instead of using the Bactrian you can use another heavy warship such as the Dreadnought, Falcon or Leviathan. I recommend the Dreadnought if you have access to it.) After purchasing, modify it as such:

All of these modifications can be done on the same planet you purchase the Bactrian.

At this point you should have enough knowledge of mining to know how to find a good system. Some great ways of identifying a good system to mine are:
-Finding a rare ore such as uranium or platinum
-Finding a system with a whole lot of asteroids
-Systems with a planet to land on are optimal

But honestly, once you have the Bactrian, you can just get some smaller ships to gather around your flagship and fly around guns blazing. You'll still end up with a decent amount of ore. Don't worry about collecting bad ores like silicon anymore. You have a ton of cargo space and don't need to worry about filling it up with bad ore. You can buy some pretty good ships from aliens, which will be a great way to improve mining. If you want to buy your own ship with your own customizations here are some tips for a good mining ship:
-At least 100 cargo space
-At least 8 weapons (Turrets and guns combined)
-When customizing, remember you don't need great shields on a mining ship. Trade your shield generator for more outfit space if needed.
-Enough power to use thrusting, turning and weapons without draining too fast
-As with all ships, make sure you keep the heat low

Ascella ins't the only system with good mining prospects. Try these ones out too!
-Phecda - High concentration of titanium and lead

This section is just some general tricks I've thought of while playing, as well as writing this guide.

Early Game:
For new players, you can speed up the game to 3x speed! (I'm pretty sure it's 3x but don't quote me on that.) The default key for this is caps lock.

Mid Game:
During the time you're grinding with the Hauler, it can be useful to replace your two Beam Lasers (In the guns slots, not the turrets) with Heavy Beam Lasers. You may need to take breaks to regenerate energy, but these will help you cut even Uranium asteroids in less than 3 seconds! (Assuming you're using 3x speed) You'll need to remove your fuel pods for this, but they aren't very important in the long run.

Late Game:
If you buy a few smaller ships (Interceptors like Sparrows or Furies) you can use them as pick up units by using the 'gather around flagship' command. (Default G.) These ships have a small amount of cargo but upon pressing G they will fly back and forth over your flagship, and can pick up ores for you. Just be carfeul of pirates! If pirates become a problem blowing up your small ships, consider using Light Warships instead.
Big thanks to esotericist for finding this out, but you can right click a location on-screen while flying your ship to have your fleet gather there. This is also a very efficient method for picking up ores that you might not be able to catch!