For those of you completely new to the game, it can be very overwhelming to try to figure out what to do to help you progress as a miner. I'll go over some of the first steps I recommend working on to improve your mining. Because I'm Gallente myself, I'll be writing this guide towards the perspective of a Gallente miner.
So you have a Velator and a civilian mining laser. Where do you start? First of all, make sure to do all the tutorial missions. These will give you some valuable experience and some handy pocket change. If you immediately start training up the Gallente Frigate skills, you will soon be able to fly a Navitas frigate. This is FAR better than the velator, and will give you some serious mining bonuses. For each Gallente Frigate level you train, your mining yield will go up by 20% (this is HUGE).
Next you will probably want to start upgrading your ship. Equipping better mining lasers will drastically boost the amount of ore you can mine. Boosting your cargo capacity will allow you to haul more ore with every trip, and cut down on the amount of time you spend warping back and forth between the station and the asteroid field. Cargo expanders are reasonably cheap and will give you about 30% more cargo room when you have 2 in your Navitas. You will also want to go for Miner 1 mining lasers. You might be able to fit a miner 1 and a miner 2 if your ship has enough CPU. Miners use up lots of CPU so that will usually limit the amount of mining turrets you can equip at any given time.
A mining drone may help you to mine faster too. Put one in your drone bay and set it to start mining and you will get about 15M3 more every minute. It's a tiny difference but everything helps.
Now just find a good spot to mine some decent material.
While you're making money and saving it up, start training the skills you will need to get a cruiser. This will greatly increase your mining efficiency and ship strength. Once you have a cruiser, you should be safe to mine in lower security areas down to about .5 sec. You will need Gallente Frigate 4, Spaceship Command 3, and Cruiser 1 or higher depending on the ship you want.