First of all, to make the difference between 'under' and 'below'. Both of these words can mean 'in a lower position than', so there's a sense in which they mean the same thing. But we use them sometimes in different circumstances, for example, if you're talking about something being covered by something, we use 'under'. So, 'I hid the key under a rock'. Or, 'officials said there was nothing under President Bush's jacket'.
You use 'below' when you're talking about something that's not physically immediately under, or not necessarily immediately under. So you say, 'below the surface of the water'. That might be anywhere below the surface of the water, not necessarily just touching it. Or, 'twenty miles below the earth's surface', definitely not immediately under it. And, by extension, we say things like, 'below the poverty line'.
We also use 'under' when we're talking about 'younger than' or 'less than'. So, 'under a dozen times', 'under the age of ten'. Whereas we use 'below', if we're visualising a kind of vertical scale. So, 'below sea level', 'below average', 'an IQ below 80', 'radio waves below 22 kHz'.
There are a number of fixed expressions, so, for example, a lot of expressions about what's happening while something else is going on, or because of certain conditions, or controlled by something or someone. So we say, 'under construction', 'under fire', 'under attack', 'under arrest', 'under these conditions', 'under scrutiny', 'under pressure', 'under the Ceausescu regime'. All of those form a kind of a family.
So what about 'beneath'? Well, 'beneath' is basically more literal, or formal, and we use it in many of the same senses. But there are lots of fixed phrases, and so what you want to do is just read a lot and note when one is used and when the other is used. I hope those will give you some general guide lines, and that you'll enjoy keeping learning about these
three fascinating words