I always shake hands or- if I know them either or a hug. Man or woman.
I’ll stop short at hugging, except with a very few close people*, but I do like a handshake. It sets a tone of commitment to the meeting, which I think we all value.
In the business context, a handshake and then just my name and company. If I already know them, then a handshake and something along the lines of “great to see you – how are you doing?”.
In the wider social context, then a handshake and a “great to meet you” or similar (depending on how great it is), or something more suited to the occasion (I wouldn’t say that to a bathroom salesman, for example).
I like “how do you do”, but don’t often deploy it these days. I think I’ll make an effort to try it out more often and see how it goes.