The lure of mining is one of the cardinal "shining paths" to wealth in EVE Online, and mining, done right, is geared towards making you the most money possible in the least amount of time in the EVE Online shared game. EVE Online seeds new resources to be mined through the 5,000 star cluster every time new players make permanent accounts, so mining starts with prospecting – finding what needs to be done. An EVE Online mining guide will walk you through the basic process.
The basics of an EVE Online mining guide include watching the fringes of the universe; this is the place where EVE Online seeds new resources when new accounts are made; there's some weighting of new materials in more established areas, but the feel of EVE Online, that of a gold rush town teetering on the bringing of incivility at the edge of civilization is important. Consequently, most good EVE Online mining guides will tell you the same thing – look to the fringes, and look for places that have control ratings of 0.3 to 0.8 or so.
Once you've found a spot to mine, there's the temptation to process the ore yourself. While this is tempting, it's usually not worth it, because of how EVE Online does skills – getting both mining and processing to respectable levels means you're cutting out a lot of other things that are fun to do as well, and the processing gear is more expensive than the mining gear. Most mining conglomerates doing EVE online mining guides recommend that you do it as a group activity, with two or three miners per processor, and often with a couple of dedicated combat types to do guard and patrol duty.
Always check for piracy and similar activities, whether generated by the game or, more dangerously, by other players. A useful technique for avoiding pirates (and not leading them to your mining site) is to bookmark your asteroid, so you end up arriving right at it from the jump point.
The ultimate goal of mining is to maximize the amount of ISK you get per unit of time. The easiest way to do this is with a mining macro, but most dedicated (and worthwhile) EVE Online mining guides will advise against this. First of all, macros are easily detected. Second, they get your account nuked and your assets vaporized if you get caught. Instead, focus on playing the market – with unprocessed ores, there's going to be a market price, and there is not much cost-benefit in trying to get the best price – far better to sell your hold worth of stuff now, and get back to your claim as quickly as possible!