A few quick things that will help you get started quicker and easier:

Mining - When you're trying to mine an asteroid with the mining/tractor beam, you have to be close enough to see the beam burning the surface of the rock. This is usually ~100m from the object. When you're first starting out, unless you have a mission to get Metal Alloys, discard them from your holds so you only mine platinum and diamonds. These will get you more money no matter where you go. The beam will get you hydrogen from nebulae (or fuel if you're using a fuel converter), water from ringed planets' rings, oxygen from planet atmospheres and biological materials from plants/etc on planet surfaces.

Navigation - You'll find pretty quickly that using the Nav screen/map is essential, as it should be. Things like stations and jump gates are shown on the map. You might be tempted to just get your cursor on top of the station/jumpgate and hit your autopilot. While that will work, once you get within the vicinity of the destination, you're left flying manually to dock or jump. If, however, you RIGHT CLICK the station/jump gate in the nav map, it will 'auto-center' it and when you jump there, you should fly straight in to the dock or through the jump gate.

Information - When you're mining and/or trading, hit your inventory (F3) and check the News fairly often. It will show you what all commodities are selling for in that area at that time. Keep in mind that you might see that diamonds are worth more than platinum and dump all your platinum, only to find that the price of diamonds tanked before you got back to the station to sell it. Not a lot you can do about this other than waiting for prices to become favorable again or moving on and selling somewhere else.

The Internet - While these few tips are what I consider must-have information, there's tons of stuff out there in the intertubes, quite a lot of it really good. I urge you to take advantage of this situation. :)