Hmm, in my opinion there are always several reasons (which was typic in my protein physics course, lol).
You are on to some of it. The hydrogen bonds make most secondary structures energetically stable. This thing about forming structures, is almost always about energy.
But what defines the structure depends on the amino acids and their structure. I've completely forgot the names (lol), but you can most surely find a long list, that will tell you, which amino acids favors beta sheets, and which favors alpha helices.
Some specific amino acids are only found in the beginning or the end of the secondary structure, because it may be good at starting the bend, which leads to the helix structure. Well you hopefully relate all this to some courses?
But overall it may be a quite short answer. It is simply energetically favorable for a sequence of amino acids to form secondary structures. Which structure it forms, depends on the individual amino acids, their location with respect to each other, and of course the amount of amino acids in the sequence :-)
You also mention something about hydrophobic amino acids, which is somewhat correct, however, it is more often associated with the tertiary stucture, as it is here the different secondary structures are positioned or even bended in an energetically favorable manner, which for instance can protect the hydrophobic proteins from water. Well you definitely need illustrations to clearly understand this, I guess.