By flying into rising air: if you fly into rising air, it will carry you up with it.
The most common way is to find a hill or a cliff that faces into the wind: when the wind blows up the slope, it is deflected upwards. By flying in the air that is deflected upwards, you can stay up indefinitely. This is called ridge soaring. To stay in the rising air, you have to keep flying to-and-fro accross the hill to keep in front of the slope. If you ever saw a seagull flying in front of a cliff without flapping it's wings, this is what it was doing.
Another common way to go up is to fly into a thermal on a sunny day. A thermal is a column of warm, rising, air. It is created when air is heated by ground tht has warmed in the sun. The heated air often bubbles up thousands of feet into the heavens. If you fly into it, it will carry you up with it. This sort of flying is called thermalling. To stay in the column of rising air, you have to fly in a circle. If you ever saw a large bird circling up without flapping it's wings, this is what it was doing.
Now you understand how you can really fly like a soaring bird, such as a seagul or an eagle !
That depends on the conditions and your skill: exceptionally, it is possible to go over twenty thousand feet, where you will need oxygen. It is more common to rise a hundred feet or so when you are ridge soaring (see the question ‘How do you stay up?’ above) or a few thousand feet when you are thermalling (again, see the question ‘How do you stay up?’ above).
Not particularly, if you've been trained. The reputation for danger which hang gliding had in the early days was due to the unsafe design of early hang gliders, and a lack of training. It is probably more dangerous than driving a car but arguably less dangerous than smoking. As with driving a car, you can make it more or less dangerous depending on how you fly.
The opposite of 'How do you stay up?': by flying into air that isn't rising. In the beginning, it is all too easy to come down because the rising air is hard to find. With a little experience, it becomes easier to find where the air is rising and where it is not, and so to choose whether to go up or come down.
You normally go to a training school and have lessons until you receive a formal qualification. It's a bit like learning to drive a car. There's no need to be especially fit or young. For more information, click on How to Learn at the top of this page.
A rough guide is that it is as cheap or cheaper to learn to fly a paraglider or a hang glider than it is to learn to drive a car. Once you're trained, the cost of a decent second hand hang glider or paraglider and accessories is probably less than a cheap second hand car. Thereafter, all you pay for is the petrol to get to the flying site.
I like to put it this way: it is cheaper than playing darts down the pub, if you take incidentals into account.
Yes !
Hang gliders and paragliders spend most of the time pointing into wind when they're near the ground. This makes spectators wonder if you can steer them. Yes, you can. You can turn right and left just like a car. If you couldn’t steer, you couldn’t, for instance, choose where to land. This question often causes amusement amongst pilots: Imagine driving to a friends house, and when you arrive, your friend looks at the car and asks if you can steer it.