Mining is one of the most profitable skills in Runescape, especially if you can put it together with Smithing, but that requires a lot of work. The easiest is to pick one and focus, since you can never mine enough to smith. No matter what you choose to do in Runescape, you will want to at least get your mining up to 15, since that can provide you with some basic ores for other things, such as crafting.
To start mining, you need a pick axe. A simple bronze pick axe will do for now, until you have a higher mining. Once you have a pick (you should have from tutorial island), go to a mine. You can find a mine by looking on the world map. Simply (left) click on the rock you want to mine. This may not be a rock you can mine yet, if you repeatedly get no ore. Compare what the rock looks like to the chart below. Note: you can also right-click on a rock, and select prospect to see which ore is in the rock, if you can not tell after a few tries, it is probably out of your mining range.
When you start you can only mine copper and tin, but with a little work you can soon start to be mining coal and making the money.
Do not try to sell any of your copper or tin ore, most players will not buy, and it will waste time you can otherwise be spending mining. If you do not plan to smith your copper or tin into bronze, my recommendation would be to just drop the ore, since it will do no good taking up bank space.
One of the most popular mines is the Dwarven mine, located under Ice Mountain, and another entrance from Falador. It has every type of ore in it (except rune), and it is close to a bank for convenience.
This makes it one of the most crowded mines, which can make it hard to get ore with so many competitors.
Many places in Runescape have monsters, as you have seen, and mines are not an exception. The most common monster in mines is the scorpion (lvl 21), which has killed many miners, both strong and weak. Until you are confident in your ability to defeat a scorpion, bring protection (sword, armour), especially in case of the King Scorpion (lvl 32).