Only politicians care. Ordinary people in America don't care. Just extend your hand and offer a handshake. It's always polite.

In America, usually the person that introduces themselves begins the handshake, but there is no general rule to this, and it is not rude for a student to begin shaking a professor's hand.

It seems like in America if you want to shake someone's hand you just do put it out there and do it, it doesn't really matter on social position, you won't be looked down upon for sticking out your hand first for say a professor. I will say I always think it's a LITTLE weird when I'm introduced to a friend's friend and they shake my hand because I usually think of that as being really informal and not necessary but not in a "oh okay that guy is a total weirdo" sort of way just in a "oh okay sure". It's kind of the same reaction I get when a friends friend wants to give me a hug....a little weird but not impolite in the least. Really it's just a sign of being polite. I've never seen a poor reaction to getting or giving a handshake in any situation. I feel like it also depends on where you are, I feel like when I'm visiting the south/Midwest there's a lot more handshaking In more casual situations than up north/east.

Nah... Men don't have to wait for women to initiate a hand shake. Just shove your hand at me and I'll happily reciprocate! Here in the US, you can offer a hand shake to anyone in an introductory meeting regardless the person's status. It's only weird if you do it to random strangers. Shake away!

In American culture a handshake isn't always necessary you can just introduce yourself and smile as well. But a handshake is a sign of sincerity not disrespect regardless of your age or status (teacher-student / boss-employee)